Princeton Review Ap Calc if G(X)= 1/32 X^4

Learning Objectives

• 1.i.i Use functional notation to evaluate a function.
• i.i.ii Determine the domain and range of a function.
• 1.1.three Draw the graph of a part.
• 1.1.4 Find the zeros of a function.
• 1.1.5 Recognize a function from a table of values.
• 1.i.6 Make new functions from two or more given functions.
• 1.one.7 Describe the symmetry properties of a function.

In this section, nosotros provide a formal definition of a function and examine several ways in which functions are represented—namely, through tables, formulas, and graphs. We study formal notation and terms related to functions. We also define composition of functions and symmetry properties. Nearly of this material will be a review for you, just it serves as a handy reference to remind y'all of some of the algebraic techniques useful for working with functions.

Functions

Given two sets $A$ and $B,$ a set with elements that are ordered pairs $(\mathrm{10},y),$ where $x$ is an element of $A$ and $y$ is an chemical element of $B,$ is a relation from $A$ to $B.$ A relation from $A$ to $B$ defines a relationship between those ii sets. A role is a special type of relation in which each element of the first set is related to exactly one chemical element of the second ready. The chemical element of the first prepare is called the input ; the element of the 2nd set is called the output . Functions are used all the fourth dimension in mathematics to describe relationships between two sets. For whatsoever function, when we know the input, the output is determined, so we say that the output is a function of the input. For example, the area of a square is determined by its side length, so nosotros say that the area (the output) is a function of its side length (the input). The velocity of a ball thrown in the air can be described as a role of the amount of time the ball is in the air. The cost of mailing a package is a function of the weight of the package. Since functions have so many uses, it is important to have precise definitions and terminology to study them.

Definition

A function $f$ consists of a set of inputs, a set of outputs, and a dominion for assigning each input to exactly one output. The set of inputs is chosen the domain of the function. The set up of outputs is called the range of the function.

For instance, consider the function $f,$ where the domain is the set of all real numbers and the rule is to foursquare the input. Then, the input $\mathrm{10}=\mathrm{iii}$ is assigned to the output ${\mathrm{iii}}^2=\mathrm{ix}.$ Since every nonnegative real number has a real-value square root, every nonnegative number is an element of the range of this part. Since at that place is no real number with a foursquare that is negative, the negative real numbers are not elements of the range. Nosotros conclude that the range is the set of nonnegative real numbers.

For a general function $f$ with domain $D,$ we frequently use $x$ to denote the input and $y$ to announce the output associated with $x.$ When doing so, we refer to $\mathrm{ten}$ equally the independent variable and $y$ as the dependent variable, because information technology depends on $x.$ Using function notation, we write $y=f(x),$ and we read this equation every bit $\text{"}y$ equals $f$ of $\mathrm{10}.\text{"}$ For the squaring function described earlier, nosotros write $f(\mathrm{10})={\mathrm{ten}}^{\mathrm{two}}.$

The concept of a function can be visualized using Figure 1.ii, Figure 1.iii, and Effigy i.4.

Effigy ane.2 A role can be visualized as an input/output device.

Figure one.3 A function maps every element in the domain to exactly 1 element in the range. Although each input tin exist sent to simply one output, two unlike inputs can be sent to the aforementioned output.

Effigy 1.4 In this case, a graph of a role $f$ has a domain of $\{1,2,\mathrm{iii}\}$ and a range of $\{\mathrm{one},2\}.$ The independent variable is $x$ and the dependent variable is $y.$

We can also visualize a function by plotting points $(x,y)$ in the coordinate aeroplane where $y=f(x).$ The graph of a role is the fix of all these points. For example, consider the part $f,$ where the domain is the fix $D=\{1,\mathrm{ii},3\}$ and the rule is $f(\mathrm{ten})=3-x.$ In Figure 1.five, we plot a graph of this function.

Figure 1.five Here we run across a graph of the function $f$ with domain $\{1,2,\mathrm{three}\}$ and rule $f(\mathrm{10})=\mathrm{three}-x.$ The graph consists of the points $(x,f\left(\mathrm{ten}\right))$ for all $\mathrm{ten}$ in the domain.

Every role has a domain. However, sometimes a function is described by an equation, every bit in $f(x)=x^2,$ with no specific domain given. In this case, the domain is taken to be the set of all existent numbers $x$ for which $f(x)$ is a real number. For example, since any real number can be squared, if no other domain is specified, we consider the domain of $f(x)=x^{\mathrm{ii}}$ to be the fix of all real numbers. On the other mitt, the square root function $f(x)=\sqrt{\mathrm{10}}$ only gives a real output if $x$ is nonnegative. Therefore, the domain of the part $f\left(x\right)=\sqrt{x}$ is the set up of nonnegative real numbers, sometimes chosen the natural domain.

For the functions $f(x)=x^2$ and $f(\mathrm{ten})=\sqrt{x},$ the domains are sets with an space number of elements. Clearly we cannot list all these elements. When describing a ready with an infinite number of elements, it is often helpful to employ set-builder or interval annotation. When using set-builder annotation to draw a subset of all real numbers, denoted $ℝ,$ nosotros write

$$\left\{\mathrm{ten}\right|\mathrm{ten}\text{has some property}\}.$$

We read this equally the set of existent numbers $x$ such that $x$ has some holding. For case, if we were interested in the set of real numbers that are greater than ane merely less than v, nosotros could denote this set using set-builder notation by writing

$$\left\{x\right|1<x<5\}.$$

A ready such as this, which contains all numbers greater than $a$ and less than $b,$ tin can also be denoted using the interval annotation $(a,b).$ Therefore,

$$(\mathrm{one},\mathrm{five})=\left\{x\right|\mathrm{i}<x<\mathrm{v}\}.$$

The numbers $1$ and $\mathrm{five}$ are called the endpoints of this set. If we want to consider the gear up that includes the endpoints, we would denote this set by writing

$$[\mathrm{one},5]=\left\{x\right|\mathrm{i}\le \mathrm{10}\le \mathrm{five}\}.$$

We can use similar notation if nosotros want to include one of the endpoints, just not the other. To announce the set of nonnegative existent numbers, we would employ the set-builder annotation

$$\left\{x\right|0\le x\}.$$

The smallest number in this set up is zero, but this ready does non take a largest number. Using interval annotation, we would use the symbol $\infty ,$ which refers to positive infinity, and we would write the set as

$$[0,\infty )=\left\{x\right|0\le x\}.$$

Information technology is important to note that $\infty$ is not a real number. It is used symbolically here to indicate that this prepare includes all real numbers greater than or equal to nil. Similarly, if nosotros wanted to describe the set up of all nonpositive numbers, nosotros could write

$$(\text{−}\infty ,0]=\left\{x\right|x\le 0\}.$$

Here, the notation $\text{−}\infty$ refers to negative infinity, and information technology indicates that we are including all numbers less than or equal to goose egg, no matter how pocket-size. The fix

$$(\text{−}\infty ,\infty )=\left\{x\right|\mathrm{ten}\text{is whatsoever real number}\}$$

refers to the ready of all real numbers.

Some functions are defined using dissimilar equations for different parts of their domain. These types of functions are known as piecewise-defined functions . For case, suppose nosotros want to ascertain a role $f$ with a domain that is the ready of all real numbers such that $f(x)=3x+\mathrm{i}$ for $\mathrm{ten}\ge \mathrm{ii}$ and $f(x)=x^2$ for $x<2.$ Nosotros denote this function by writing

$$f(\mathrm{10})=\{\begin{array}{l}\mathrm{iii}\mathrm{10}+1x\ge \mathrm{two}\\ x^{\mathrm{ii}}x<\mathrm{two}\end{array}.$$

When evaluating this function for an input $\mathrm{ten},$ the equation to use depends on whether $\mathrm{10}\ge 2$ or $x<2.$ For example, since $5>2,$ we use the fact that $f(\mathrm{ten})=3x+1$ for $\mathrm{10}\ge 2$ and see that $f(5)=3(\mathrm{five})+1=16.$ On the other hand, for $\mathrm{10}=\mathrm{-1},$ we use the fact that $f(\mathrm{ten})=x^{\mathrm{two}}$ for $x<2$ and see that $f(\mathrm{-1})=\mathrm{i}.$

Instance 1.one

Evaluating Functions

For the function $f\left(x\right)=3x^2+2x-1,$ evaluate

1. $f\left(\mathrm{-2}\right)$
2. $f\left(\sqrt{\mathrm{ii}}\right)$
3. $f(a+h)$

Checkpoint 1.1

For $f(x)=x^{\mathrm{two}}-3x+5,$ evaluate $f(1)$ and $f(a+h).$

Example 1.2

Finding Domain and Range

For each of the following functions, determine the i. domain and ii. range.

1. $f(x)={(x-4)}^{\mathrm{two}}+5$
2. $f(x)=\sqrt{3x+\mathrm{ii}}-1$
3. $f(x)=\frac{3}{x-2}$

Checkpoint ane.2

Observe the domain and range for $f\left(x\right)=\sqrt{4-2\mathrm{10}}+\mathrm{v}.$

Representing Functions

Typically, a role is represented using 1 or more than of the following tools:

• A table
• A graph
• A formula

We can identify a part in each form, but nosotros can besides use them together. For case, we can plot on a graph the values from a table or create a table from a formula.

Tables

Functions described using a table of values arise often in real-world applications. Consider the following simple example. We can describe temperature on a given twenty-four hours as a part of time of twenty-four hour period. Suppose nosotros record the temperature every hour for a 24-hour period starting at midnight. We let our input variable $x$ exist the time after midnight, measured in hours, and the output variable $y$ exist the temperature $x$ hours later midnight, measured in degrees Fahrenheit. We record our data in Table 1.1.

Hours afterwards Midnight	Temperature $(\text{°}F)$	Hours afterwards Midnight	Temperature $(\text{°}F)$
0	58	12	84
1	54	13	85
two	53	14	85
3	52	15	83
4	52	16	82
five	55	17	80
half dozen	60	18	77
7	64	19	74
viii	72	20	69
9	75	21	65
10	78	22	60
11	eighty	23	58

Table 1.1 Temperature as a Function of Fourth dimension of Solar day

We can encounter from the table that temperature is a part of time, and the temperature decreases, and then increases, then decreases again. Yet, nosotros cannot get a articulate picture of the behavior of the function without graphing information technology.

Graphs

Given a function $f$ described by a tabular array, we can provide a visual pic of the role in the form of a graph. Graphing the temperatures listed in Table 1.1 can give us a better idea of their fluctuation throughout the day. Figure ane.six shows the plot of the temperature function.

Figure 1.half-dozen The graph of the information from Table 1.1 shows temperature as a role of time.

From the points plotted on the graph in Figure ane.6, nosotros tin can visualize the full general shape of the graph. It is ofttimes useful to connect the dots in the graph, which represent the information from the table. In this instance, although we cannot make whatsoever definitive conclusion regarding what the temperature was at whatever fourth dimension for which the temperature was not recorded, given the number of data points nerveless and the pattern in these points, it is reasonable to suspect that the temperatures at other times followed a similar pattern, as nosotros can see in Effigy 1.7.

Figure 1.7 Connecting the dots in Figure 1.6 shows the general pattern of the data.

Algebraic Formulas

Sometimes we are not given the values of a function in table form, rather nosotros are given the values in an explicit formula. Formulas arise in many applications. For instance, the area of a circle of radius $r$ is given by the formula $A\left(r\right)=\pi r^2.$ When an object is thrown upward from the basis with an initial velocity ${\mathrm{five}}_0$ ft/due south, its height above the ground from the time information technology is thrown until it hits the ground is given by the formula $\mathrm{due\; south}\left(t\right)=\mathrm{-16}{t}^2+v_{0}t.$ When $P$ dollars are invested in an account at an annual interest rate $r$ compounded continuously, the amount of money after $t$ years is given by the formula $A\left(t\right)=P{e}^{rt}.$ Algebraic formulas are important tools to calculate function values. Often we also correspond these functions visually in graph form.

Given an algebraic formula for a role $f,$ the graph of $f$ is the fix of points $\left(x,f\left(x\right)\right),$ where $\mathrm{10}$ is in the domain of $f$ and $f(x)$ is in the range. To graph a function given by a formula, it is helpful to brainstorm by using the formula to create a table of inputs and outputs. If the domain of $f$ consists of an infinite number of values, we cannot list all of them, just because list some of the inputs and outputs tin can exist very useful, it is often a good manner to begin.

When creating a tabular array of inputs and outputs, we typically check to determine whether zippo is an output. Those values of $\mathrm{ten}$ where $f\left(x\right)=0$ are chosen the zeros of a function. For example, the zeros of $f\left(x\right)=x^2-4$ are $x=\pm 2.$ The zeros determine where the graph of $f$ intersects the $x$ -axis, which gives us more information about the shape of the graph of the function. The graph of a function may never intersect the 10-axis, or information technology may intersect multiple (or even infinitely many) times.

Another betoken of interest is the $y$ -intercept, if it exists. The $y$ -intercept is given past $\left(0,f\left(0\right)\right).$

Since a office has exactly one output for each input, the graph of a function tin have, at near, 1 $y$ -intercept. If $x=0$ is in the domain of a function $f,$ then $f$ has exactly i $y$ -intercept. If $x=0$ is not in the domain of $f,$ and then $f$ has no $y$ -intercept. Similarly, for whatever existent number $c,$ if $c$ is in the domain of $f,$ there is exactly ane output $f\left(c\right),$ and the line $x=c$ intersects the graph of $f$ exactly once. On the other hand, if $c$ is not in the domain of $f,f(c)$ is not defined and the line $x=c$ does not intersect the graph of $f.$ This belongings is summarized in the vertical line test.

Rule: Vertical Line Test

Given a part $f,$ every vertical line that may be drawn intersects the graph of $f$ no more than once. If any vertical line intersects a fix of points more than in one case, the set of points does not represent a function.

We can use this test to determine whether a set of plotted points represents the graph of a function (Figure 1.viii).

Figure 1.8 (a) The prepare of plotted points represents the graph of a part because every vertical line intersects the fix of points, at nigh, in one case. (b) The set of plotted points does not correspond the graph of a role because some vertical lines intersect the set of points more than in one case.

Instance 1.3

Finding Zeros and $y$-Intercepts of a Function

Consider the part $f\left(x\right)=\mathrm{-4}x+2.$

1. Find all zeros of $f.$
2. Discover the $y$-intercept (if whatsoever).
3. Sketch a graph of $f.$

Example 1.4

Using Zeros and $y$-Intercepts to Sketch a Graph

Consider the function $f(x)=\sqrt{x+3}+\mathrm{ane}.$

1. Detect all zeros of $f.$
2. Detect the $y$-intercept (if any).
3. Sketch a graph of $f.$

Checkpoint i.three

Find the zeros of $f\left(x\right)={\mathrm{ten}}^3-5x^2+6x.$

Instance 1.5

Finding the Acme of a Free-Falling Object

If a brawl is dropped from a height of $100$ ft, its height $s$ at time $t$ is given by the function $s\left(t\right)=\mathrm{-16}{t}^2+100,$ where $s$ is measured in feet and $t$ is measured in seconds. The domain is restricted to the interval $[0,c],$ where $t=0$ is the time when the brawl is dropped and $t=c$ is the time when the ball hits the footing.

1. Create a table showing the height $\mathrm{south}(t)$ when $t=0,0.5,1,\mathrm{one.5},\mathrm{ii},\text{and}2.5.$ Using the data from the table, make up one's mind the domain for this function. That is, notice the fourth dimension $c$ when the brawl hits the ground.
2. Sketch a graph of $\mathrm{due\; south}.$

Note that for this function and the function $f\left(x\right)=\mathrm{-iv}x+\mathrm{two}$ graphed in Figure 1.9, the values of $f(x)$ are getting smaller as $\mathrm{ten}$ is getting larger. A function with this property is said to exist decreasing. On the other hand, for the office $f\left(\mathrm{ten}\right)=\sqrt{x+\mathrm{iii}}+1$ graphed in Figure i.10, the values of $f(\mathrm{10})$ are getting larger as the values of $\mathrm{ten}$ are getting larger. A function with this belongings is said to be increasing. Information technology is important to annotation, however, that a office tin can be increasing on some interval or intervals and decreasing over a different interval or intervals. For case, using our temperature function in Effigy ane.half dozen, we can see that the part is decreasing on the interval $(0,\mathrm{four}),$ increasing on the interval $(\mathrm{iv},14),$ and so decreasing on the interval $\left(14,23\right).$ Nosotros make the idea of a function increasing or decreasing over a particular interval more than precise in the next definition.

Definition

We say that a function $f$ is increasing on the interval $I$ if for all ${\mathrm{10}}_{\mathrm{one}},x_2\in I,$

$$f(x_1)\le f(x_2)\text{when}{x}_{\mathrm{ane}}<{\mathrm{ten}}_2.$$

We say $f$ is strictly increasing on the interval $I$ if for all ${\mathrm{10}}_1,{\mathrm{ten}}_{\mathrm{ii}}\in I,$

$$f({\mathrm{ten}}_{\mathrm{one}})<f(x_{\mathrm{ii}})\text{when}{x}_1<x_2.$$

We say that a function $f$ is decreasing on the interval $I$ if for all ${\mathrm{ten}}_1,x_{\mathrm{two}}\in I,$

$$f(x_1)\ge f(x_2)\text{if}{x}_{\mathrm{i}}<x_{\mathrm{two}}.$$

We say that a function $f$ is strictly decreasing on the interval $I$ if for all ${\mathrm{10}}_1,x_{\mathrm{ii}}\in I,$

$$f(x_{\mathrm{one}})>f(x_2)\text{if}{x}_{\mathrm{ane}}<x_{\mathrm{two}}.$$

For example, the office $f(\mathrm{10})=3x$ is increasing on the interval $(\text{−}\infty ,\infty )$ because $3{\mathrm{10}}_{\mathrm{one}}<3x_2$ whenever ${\mathrm{10}}_1<x_2.$ On the other hand, the function $f\left(x\right)=\text{−}{x}^3$ is decreasing on the interval $(\text{−}\infty ,\infty )$ because $\text{−}{\mathrm{ten}}_1^3>-{\mathrm{ten}}_2^3$ whenever $x_1<{\mathrm{10}}_2$ (Effigy one.11).

Figure 1.xi (a) The role $f(\mathrm{10})=3\mathrm{ten}$ is increasing on the interval $(\text{−}\infty ,\infty ).$ (b) The function $f\left(\mathrm{ten}\right)=\text{−}{x}^3$ is decreasing on the interval $(\text{−}\infty ,\infty ).$

Combining Functions

Now that we accept reviewed the basic characteristics of functions, we can encounter what happens to these backdrop when we combine functions in unlike ways, using basic mathematical operations to create new functions. For example, if the cost for a company to manufacture $x$ items is described past the role $C(\mathrm{10})$ and the revenue created by the auction of $x$ items is described by the office $R\left(\mathrm{ten}\right),$ then the profit on the manufacture and sale of $x$ items is defined as $P\left(\mathrm{ten}\right)=R\left(x\right)-C\left(\mathrm{10}\right).$ Using the departure betwixt two functions, we created a new function.

Alternatively, nosotros can create a new role by composing two functions. For example, given the functions $f\left(x\right)=x^2$ and $g\left(\mathrm{ten}\right)=\mathrm{three}x+1,$ the blended part $f\circ \mathrm{yard}$ is defined such that

$$\left(f\circ g\right)\left(x\right)=f\left(\mathrm{grand}\left(x\right)\right)={\left(g\left(x\right)\right)}^2={\left(3\mathrm{10}+1\right)}^{\mathrm{ii}}.$$

The composite function $\mathrm{one\; thousand}\circ f$ is defined such that

$$\left(g\circ f\right)\left(x\right)=k\left(f\left(x\right)\right)=3f\left(x\right)+1=3x^2+1.$$

Note that these 2 new functions are dissimilar from each other.

Combining Functions with Mathematical Operators

To combine functions using mathematical operators, nosotros simply write the functions with the operator and simplify. Given two functions $f$ and $g,$ nosotros tin define 4 new functions:

$$\begin{array}{cccc}\left(f+g\right)\left(\mathrm{10}\right)=f\left(x\right)+g\left(x\right)\hfill & & & \mathrm{Southward}um\hfill \\ \left(f-g\right)\left(x\right)=f\left(x\right)-k\left(x\right)\hfill & & & \mathit{\text{Departure}}\hfill \\ \left(f·\mathrm{chiliad}\right)\left(x\right)=f\left(x\right)\mathrm{thousand}\left(\mathrm{10}\right)\hfill & & & Product\hfill \\ \left(\frac{f}{g}\right)\left(\mathrm{10}\right)=\frac{f\left(x\right)}{g\left(x\right)}\text{for}k\left(x\right)\ne 0\hfill & & & Quotient\hfill \end{array}$$

Example 1.vi

Combining Functions Using Mathematical Operations

Given the functions $f\left(x\right)=2\mathrm{ten}-3$ and $\mathrm{chiliad}\left(\mathrm{10}\right)={\mathrm{ten}}^{\mathrm{two}}-1,$ find each of the post-obit functions and state its domain.

1. $(f+g)(x)$
2. $(f-g)(x)$
3. $(f·k)(\mathrm{ten})$
4. $\left(\frac{f}{g}\right)\left(\mathrm{ten}\right)$

Checkpoint 1.four

For $f\left(x\right)=x^{\mathrm{ii}}+\mathrm{three}$ and $\mathrm{thousand}\left(x\right)=2x-\mathrm{v},$ find $\left(f\text{/}g\right)\left(x\right)$ and state its domain.

Function Limerick

When nosotros etch functions, we take a function of a part. For example, suppose the temperature $T$ on a given twenty-four hour period is described equally a role of time $t$ (measured in hours after midnight) as in Table ane.one. Suppose the cost $C,$ to rut or absurd a building for one hour, can be described as a function of the temperature $T.$ Combining these two functions, nosotros can describe the cost of heating or cooling a building as a part of fourth dimension past evaluating $C\left(T\left(t\right)\right).$ Nosotros have defined a new function, denoted $C\circ T,$ which is defined such that $\left(C\circ T\right)\left(t\right)=C(T\left(t\right))$ for all $t$ in the domain of $T.$ This new function is called a composite function. We notation that since cost is a part of temperature and temperature is a part of time, it makes sense to define this new office $(C\circ T)(t).$ It does not make sense to consider $(T\circ C)(t),$ because temperature is non a office of price.

Definition

Consider the function $f$ with domain $A$ and range $B,$ and the office $g$ with domain $D$ and range $E.$ If $B$ is a subset of $D,$ then the composite function $(g\circ f)(\mathrm{ten})$ is the function with domain $A$ such that

$$\left(m\circ f\right)\left(\mathrm{ten}\right)=\mathrm{chiliad}\left(f\left(\mathrm{10}\right)\right).$$

(1.1)

A composite role $g\circ f$ can be viewed in two steps. Start, the part $f$ maps each input $x$ in the domain of $f$ to its output $f(x)$ in the range of $f.$ Second, since the range of $f$ is a subset of the domain of $g,$ the output $f(\mathrm{10})$ is an element in the domain of $\mathrm{yard},$ and therefore information technology is mapped to an output $g\left(f\left(\mathrm{10}\right)\right)$ in the range of $\mathrm{yard}.$ In Effigy ane.12, we encounter a visual image of a composite function.

Figure i.12 For the composite office $g\circ f,$ we have $\left(m\circ f\right)\left(\mathrm{i}\right)=4,\left(\mathrm{yard}\circ f\right)\left(\mathrm{ii}\right)=\mathrm{five},$ and $\left(g\circ f\right)\left(\mathrm{iii}\right)=4.$

Example 1.7

Compositions of Functions Divers past Formulas

Consider the functions $f(x)=x^2+\mathrm{one}$ and $k(x)=\mathrm{i}\text{/}x.$

1. Detect $(g\circ f)(\mathrm{ten})$ and land its domain and range.
2. Evaluate $(g\circ f)(\mathrm{four}),(m\circ f)(\mathrm{-1}\text{/}2).$
3. Find $(f\circ g)(x)$ and country its domain and range.
4. Evaluate $(f\circ g)(4),(f\circ \mathrm{thou})(\mathrm{-ane}\text{/}2).$

In Example 1.seven, we can see that $\left(f\circ g\right)\left(\mathrm{ten}\right)\ne \left(g\circ f\right)\left(\mathrm{ten}\right).$ This tells u.s.a., in full general terms, that the order in which we etch functions matters.

Checkpoint one.v

Allow $f\left(\mathrm{10}\right)=2-5x.$ Let $\mathrm{one\; thousand}\left(\mathrm{ten}\right)=\sqrt{x}.$ Find $\left(f\circ g\right)\left(x\right).$

Example i.8

Composition of Functions Divers by Tables

Consider the functions $f$ and $g$ described by Table 1.4 and Table 1.five.

$\mathit{\text{x}}$	$\mathrm{-3}$	$\mathrm{-2}$	$\mathrm{-ane}$	0	1	ii	iii	4
$\mathit{\text{f}}\mathbf{(}\mathit{\text{x}}\mathbf{)}$	0	iv	2	iv	$\mathrm{-2}$	0	$\mathrm{-2}$	four

Table 1.four

$\mathit{\text{ten}}$	$\mathrm{-four}$	$\mathrm{-2}$	0	two	4
$\mathit{\text{k}}\mathbf{(}\mathit{\text{ten}}\mathbf{)}$	one	0	3	0	5

Tabular array 1.5

1. Evaluate $(\mathrm{chiliad}\circ f)(3),\left(g\circ f\right)\left(0\right).$
2. State the domain and range of $\left(g\circ f\right)\left(x\right).$
3. Evaluate $(f\circ f)(3),\left(f\circ f\right)\left(\mathrm{one}\right).$
4. State the domain and range of $\left(f\circ f\right)\left(x\right).$

Case 1.9

Awarding Involving a Composite Part

A store is ad a sale of $20\%$ off all trade. Caroline has a coupon that entitles her to an additional $\mathrm{fifteen}\%$ off any item, including auction merchandise. If Caroline decides to purchase an item with an original price of $x$ dollars, how much will she cease up paying if she applies her coupon to the auction price? Solve this trouble by using a composite function.

Checkpoint 1.6

If items are on sale for $10\%$ off their original toll, and a customer has a coupon for an additional $\mathrm{thirty}\%$ off, what volition be the final price for an item that is originally $x$ dollars, after applying the coupon to the sale price?

Symmetry of Functions

The graphs of sure functions take symmetry properties that help us sympathize the office and the shape of its graph. For instance, consider the function $f(x)={\mathrm{ten}}^4-2{\mathrm{10}}^{\mathrm{two}}-3$ shown in Figure 1.xiii(a). If we have the part of the curve that lies to the right of the y-axis and flip it over the y-axis, it lays exactly on peak of the curve to the left of the y-axis. In this case, nosotros say the function has symmetry about the y-axis. On the other manus, consider the function $f(\mathrm{ten})={\mathrm{10}}^3-4x$ shown in Figure 1.13(b). If we take the graph and rotate information technology $180\text{°}$ well-nigh the origin, the new graph volition look exactly the same. In this case, we say the function has symmetry about the origin.

Figure one.13 (a) A graph that is symmetric about the $y$-centrality. (b) A graph that is symmetric about the origin.

If nosotros are given the graph of a office, it is like shooting fish in a barrel to come across whether the graph has one of these symmetry properties. But without a graph, how tin nosotros determine algebraically whether a function $f$ has symmetry? Looking at Effigy 1.14 once again, nosotros meet that since $f$ is symmetric about the $y$ -axis, if the point $(x,y)$ is on the graph, the point $(\text{−}x,y)$ is on the graph. In other words, $f\left(\text{−}x\right)=f\left(x\right).$ If a function $f$ has this property, nosotros say $f$ is an even function, which has symmetry about the y-axis. For instance, $f(x)=x^{\mathrm{two}}$ is fifty-fifty because

$$f\left(\text{−}x\right)={\left(\text{−}x\right)}^2=x^2=f\left(x\right).$$

In contrast, looking at Figure 1.14 over again, if a part $f$ is symmetric virtually the origin, then whenever the point $(x,y)$ is on the graph, the indicate $\left(\text{−}x,\text{−}y\right)$ is also on the graph. In other words, $f(\text{−}x)=\text{−}f(x).$ If $f$ has this belongings, nosotros say $f$ is an odd role, which has symmetry almost the origin. For case, $f(\mathrm{10})=x^3$ is odd considering

$$f(\text{−}x)={(\text{−}\mathrm{10})}^3=\text{−}{x}^{\mathrm{three}}=\text{−}f(\mathrm{ten}).$$

Definition

If $f(x)=f(\text{−}x)$ for all $x$ in the domain of $f,$ then $f$ is an fifty-fifty function. An fifty-fifty role is symmetric nearly the y-axis.

If $f(\text{−}x)=\text{−}f(\mathrm{10})$ for all $x$ in the domain of $f,$ then $f$ is an odd function. An odd function is symmetric about the origin.

Case 1.ten

Even and Odd Functions

Make up one's mind whether each of the post-obit functions is even, odd, or neither.

1. $f\left(x\right)=\mathrm{-5}{\mathrm{10}}^4+\mathrm{seven}{x}^{\mathrm{ii}}-2$
2. $f(x)=2{\mathrm{10}}^5-4x+5$
3. $f(\mathrm{10})=\frac{\mathrm{iii}x}{x^2+\mathrm{ane}}$

Checkpoint one.7

Make up one's mind whether $f\left(x\right)=4{\mathrm{ten}}^3-5x$ is even, odd, or neither.

One symmetric function that arises frequently is the accented value role, written equally $\left|\mathrm{10}\right|.$ The accented value part is divers as

$$f\left(x\right)=\{\begin{array}{c}\text{−}x,\mathrm{10}<0\\ \mathrm{10},x\ge 0\end{array}.$$

(1.2)

Some students describe this office past stating that information technology "makes everything positive." By the definition of the absolute value office, we see that if $x<0,$ and then $\left|x\right|=\text{−}x>0,$ and if $\mathrm{10}>0,$ then $\left|x\right|=x>0.$ All the same, for $\mathrm{ten}=0,\left|x\right|=0.$ Therefore, it is more than accurate to say that for all nonzero inputs, the output is positive, but if $x=0,$ the output $\left|\mathrm{10}\right|=0.$ We conclude that the range of the accented value office is $\left\{y\right|y\ge 0\}.$ In Effigy 1.14, we see that the accented value function is symmetric about the y-axis and is therefore an even role.

Figure ane.fourteen The graph of $f(x)=\left|x\right|$ is symmetric well-nigh the $y$-axis.

Example 1.11

Working with the Absolute Value Function

Notice the domain and range of the function $f(\mathrm{10})=\mathrm{ii}|\mathrm{10}-3|+\mathrm{four}.$

Checkpoint 1.8

For the office $f(x)=|x+\mathrm{two}|-4,$ observe the domain and range.

Section one.1 Exercises

For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a role.

1 .

$x$	$y$	$x$	$y$
−3	9	one	1
−2	four	two	4
−1	1	3	9
0	0

two .

$x$	$y$	$x$	$y$
−3	−2	1	ane
−ii	−8	2	8
−1	−1	iii	−2
0	0

3 .

$x$	$y$	$x$	$y$
i	−three	1	ane
ii	−2	2	2
3	−i	3	3
0	0

4 .

$x$	$y$	$x$	$y$
1	ane	5	ane
ii	1	6	ane
3	one	7	1
4	one

5 .

$\mathrm{10}$	$y$	$x$	$y$
3	3	15	1
v	2	21	2
viii	one	33	3
ten	0

6 .

$x$	$y$	$x$	$y$
−7	11	i	−2
−two	5	3	four
−2	ane	6	11
0	−i

For the following exercises, find the values for each role, if they exist, and so simplify.

a. $f(0)$ b. $f(1)$ c. $f(\mathrm{three})$ d. $f(\text{−}x)$ eastward. $f(a)$ f. $f(a+h)$

seven .

$f\left(\mathrm{10}\right)=5x-\mathrm{two}$

8 .

$f\left(\mathrm{ten}\right)=\mathrm{four}{x}^2-\mathrm{iii}x+1$

ix .

$f\left(\mathrm{ten}\right)=\frac{2}{x}$

10 .

$f\left(\mathrm{ten}\right)=\left|x-\mathrm{vii}\right|+\mathrm{eight}$

eleven .

$f\left(x\right)=\sqrt{6x+\mathrm{five}}$

12 .

$f\left(\mathrm{ten}\right)=\frac{x-2}{\mathrm{three}x+\mathrm{seven}}$

xiii .

$f\left(x\right)=9$

For the post-obit exercises, find the domain, range, and all zeros/intercepts, if whatever, of the functions.

14 .

$f\left(x\right)=\frac{x}{{\mathrm{ten}}^2-16}$

15 .

$\mathrm{yard}\left(x\right)=\sqrt{8\mathrm{ten}-1}$

16 .

$h\left(x\right)=\frac{3}{x^{\mathrm{ii}}+4}$

17 .

$f\left(\mathrm{ten}\right)=\mathrm{-1}+\sqrt{x+2}$

xviii .

$f\left(x\right)=\frac{1}{\sqrt{x-\mathrm{ix}}}$

xix .

$\mathrm{thousand}\left(x\right)=\frac{3}{\mathrm{10}-4}$

20 .

$f\left(x\right)=\mathrm{four}\left|x+5\right|$

21 .

$\mathrm{grand}\left(x\right)=\sqrt{\frac{7}{\mathrm{ten}-5}}$

For the post-obit exercises, fix upward a table to sketch the graph of each office using the following values: $\mathrm{10}=\mathrm{-three},\mathrm{-2},\mathrm{-1},0,1,\mathrm{ii},3.$

22 .

$f\left(x\right)=x^2+\mathrm{one}$

$x$	$y$	$x$	$y$
−3	x	1	2
−two	v	2	five
−ane	ii	3	ten
0	1

23 .

$f\left(\mathrm{10}\right)=3\mathrm{10}-6$

$\mathrm{10}$	$y$	$\mathrm{10}$	$y$
−three	−15	ane	−3
−2	−12	two	0
−1	−9	3	3
0	−6

24 .

$f\left(\mathrm{10}\right)=\frac{\mathrm{one}}{\mathrm{ii}}x+1$

$\mathrm{ten}$	$y$	$\mathrm{10}$	$y$
−iii	$-\frac{\mathrm{one}}{\mathrm{ii}}$	1	$\frac{3}{\mathrm{ii}}$
−ii	0	2	2
−i	$\frac{\mathrm{i}}{2}$	3	$\frac{5}{2}$
0	i

25 .

$f\left(\mathrm{10}\right)=2\left|\mathrm{ten}\right|$

$x$	$y$	$x$	$y$
−3	6	i	2
−2	4	2	4
−ane	2	iii	vi
0	0

26 .

$f\left(x\right)=\text{−}{x}^2$

$x$	$y$	$x$	$y$
−3	−9	1	−1
−2	−iv	ii	−four
−ane	−1	three	−9
0	0

27 .

$f\left(x\right)={\mathrm{ten}}^3$

$\mathrm{10}$	$y$	$\mathrm{ten}$	$y$
−3	−27	1	i
−2	−8	two	8
−1	−1	3	27
0	0

For the following exercises, use the vertical line test to decide whether each of the given graphs represents a office. Assume that a graph continues at both ends if information technology extends beyond the given grid. If the graph represents a function, then determine the following for each graph:

1. Domain and range
2. $x$-intercept, if any (estimate where necessary)
3. $y$-Intercept, if any (estimate where necessary)
4. The intervals for which the function is increasing
5. The intervals for which the role is decreasing
6. The intervals for which the function is constant
7. Symmetry about any axis and/or the origin
8. Whether the role is even, odd, or neither

28 .

30 .

32 .

34 .

For the following exercises, for each pair of functions, find a. $f+m$ b. $f-g$ c. $f·g$ d. $f\text{/}g.$ Decide the domain of each of these new functions.

36 .

$f\left(x\right)=\mathrm{iii}x+4,g\left(x\right)=x-\mathrm{two}$

37 .

$f\left(x\right)=x-8,g\left(\mathrm{10}\right)=\mathrm{five}{x}^{\mathrm{ii}}$

38 .

$f\left(x\right)=3x^{\mathrm{two}}+4\mathrm{ten}+\mathrm{i},\mathrm{chiliad}\left(\mathrm{ten}\right)=x+\mathrm{one}$

39 .

$f\left(x\right)=\mathrm{nine}-x^2,\mathrm{grand}\left(\mathrm{10}\right)=x^2-2x-3$

40 .

$f\left(x\right)=\sqrt{\mathrm{10}},k\left(\mathrm{ten}\right)=\mathrm{10}-\mathrm{ii}$

41 .

$f\left(x\right)=\mathrm{half-dozen}+\frac{\mathrm{one}}{\mathrm{10}},\mathrm{grand}\left(x\right)=\frac{1}{\mathrm{ten}}$

For the following exercises, for each pair of functions, discover a. $\left(f\circ g\right)\left(x\right)$ and b. $\left(g\circ f\right)\left(x\right)$ Simplify the results. Observe the domain of each of the results.

42 .

$f\left(x\right)=3\mathrm{ten},g\left(\mathrm{ten}\right)=x+\mathrm{v}$

43 .

$f\left(x\right)=x+\mathrm{iv},g\left(x\right)=4x-1$

44 .

$f\left(\mathrm{10}\right)=2\mathrm{10}+4,m\left(\mathrm{10}\right)=x^2-2$

45 .

$f\left(x\right)=x^2+\mathrm{vii},g\left(x\right)=x^{\mathrm{two}}-\mathrm{three}$

46 .

$f\left(\mathrm{10}\right)=\sqrt{\mathrm{ten}},g\left(x\right)=x+9$

47 .

$f\left(x\right)=\frac{3}{2\mathrm{10}+1},g\left(x\right)=\frac{\mathrm{ii}}{\mathrm{ten}}$

48 .

$f(x)=\left|\mathrm{ten}+\mathrm{i}\right|,g(x)=x^2+\mathrm{ten}-4$

49 .

The table below lists the NBA championship winners for the years 2001 to 2012.

Year	Winner
2001	LA Lakers
2002	LA Lakers
2003	San Antonio Spurs
2004	Detroit Pistons
2005	San Antonio Spurs
2006	Miami Oestrus
2007	San Antonio Spurs
2008	Boston Celtics
2009	LA Lakers
2010	LA Lakers
2011	Dallas Mavericks
2012	Miami Heat

1. Consider the relation in which the domain values are the years 2001 to 2012 and the range is the corresponding winner. Is this relation a function? Explain why or why non.
2. Consider the relation where the domain values are the winners and the range is the corresponding years. Is this relation a function? Explain why or why non.

50 .

[T] The area $A$ of a square depends on the length of the side $s.$

1. Write a function $A(s)$ for the area of a square.
2. Observe and interpret $A(\mathrm{6.v}).$
3. Find the verbal and the ii-meaning-digit approximation to the length of the sides of a foursquare with expanse 56 square units.

51 .

[T] The volume of a cube depends on the length of the sides $s.$

1. Write a part $V(s)$ for the book of a cube.
2. Observe and interpret $5(\mathrm{eleven.8}).$

52 .

[T] A rental automobile visitor rents cars for a flat fee of $20 and an hourly charge of $x.25. Therefore, the total price $C$ to rent a machine is a part of the hours $t$ the motorcar is rented plus the apartment fee.

1. Write the formula for the function that models this situation.
2. Detect the total toll to rent a car for 2 days and 7 hours.
3. Decide how long the machine was rented if the pecker is $432.73.

53 .

[T] A vehicle has a 20-gal tank and gets xv mpg. The number of miles North that can be driven depends on the corporeality of gas x in the tank.

1. Write a formula that models this situation.
2. Determine the number of miles the vehicle can travel on (i) a full tank of gas and (ii) iii/4 of a tank of gas.
3. Determine the domain and range of the office.
4. Decide how many times the driver had to stop for gas if she has driven a total of 578 mi.

54 .

[T] The volume V of a sphere depends on the length of its radius as $V=(\mathrm{iv}\text{/}\mathrm{three})\pi r^3.$ Because World is non a perfect sphere, we can use the mean radius when measuring from the center to its surface. The mean radius is the boilerplate distance from the physical heart to the surface, based on a large number of samples. Find the book of Earth with mean radius $6.371\times {10}^6$ m.

55 .

[T] A certain bacterium grows in civilisation in a round region. The radius of the circle, measured in centimeters, is given by $r\left(t\right)=\mathrm{half-dozen}-\left[5\text{/}\left(t^{\mathrm{ii}}+1\right)\right],$ where t is time measured in hours since a circle of a 1-cm radius of the bacterium was put into the culture.

1. Express the expanse of the bacteria as a function of time.
2. Notice the exact and approximate area of the bacterial civilisation in 3 hours.
3. Express the circumference of the bacteria every bit a function of time.
4. Detect the exact and estimate circumference of the bacteria in iii hours.

56 .

[T] An American tourist visits Paris and must convert U.S. dollars to Euros, which can exist done using the function $E\left(x\right)=0.79\mathrm{10},$ where x is the number of U.S. dollars and $E\left(x\right)$ is the equivalent number of Euros. Since conversion rates fluctuate, when the tourist returns to the United States 2 weeks later, the conversion from Euros to U.S. dollars is $D\left(x\right)=\mathrm{ane.245}x,$ where ten is the number of Euros and $D(\mathrm{10})$ is the equivalent number of U.Due south. dollars.

1. Notice the composite function that converts straight from U.S. dollars to U.S. dollars via Euros. Did this tourist lose value in the conversion process?
2. Use (a) to determine how many U.South. dollars the tourist would get back at the stop of her trip if she converted an actress $200 when she arrived in Paris.

57 .

[T] The manager at a skateboard shop pays his workers a monthly salary South of $750 plus a commission of $8.50 for each skateboard they sell.

1. Write a function $y=S\left(x\right)$ that models a worker'southward monthly bacon based on the number of skateboards x he or she sells.
2. Find the guess monthly bacon when a worker sells 25, forty, or 55 skateboards.
3. Apply the INTERSECT feature on a graphing figurer to make up one's mind the number of skateboards that must exist sold for a worker to earn a monthly income of $1400. (Hint: Find the intersection of the office and the line $y=1400\text{.)}$

58 .

[T] Use a graphing calculator to graph the half-circle $y=\sqrt{25-{(x-4)}^2}.$ Then, use the INTERCEPT characteristic to notice the value of both the $\mathrm{ten}$ - and $y$ -intercepts.

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Source: https://openstax.org/books/calculus-volume-1/pages/1-1-review-of-functions

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