3. Functions and Graphs

Understand and Use Piecewise Functions

Basics of Functions and Their Graphs

Finding the Domain and Range of a Piecewise Function

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. h(x) = x^4 - x²+1

b. h (-1)

In Exercises 39–50, graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = x³, g(x) = x³ +2

Find the value of the function for the given value of x. See Example 3. ƒ(x)=[[x/4]], for x=7

Use the vertical line test to identify graphs in which y is a function of x. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/08e69776c573aff0-1671047101631.jpg" alt="" />

Symmetry - A Quick Discussion for Testing if a Polynomial is Even / Odd

In Exercises 11–26, determine whether each equation defines y as a function of x. 4x = y²

Determine whether each function is even, odd, or neither. See Example 5. ƒ(x)=x^5-2x^3

Fill in the blank to correctly complete each sentence.
The y-intercept of the graph of y = -2x + 6 is ________.

Find the value of the function for the given value of x. See Example 3. ƒ(x)=[[x]], for x=-√2

In Exercises 95–96, let f and g be defined by the following table: Find √(ƒ(−1) − f(0)) – [g (2)]² + ƒ(−2) ÷ g (2) · g (−1) .

Find the value of the function for the given value of x. See Example 3. ƒ(x)=[[3-(x/2)]], for x=1

In Exercises 31–32, the domain of each piecewise function is (-∞, ∞)

(a) Graph each function.
(b) Use the graph to determine the function’s range.

Domain and Range in Set Notation Tutorial

Let ƒ(x) = -2x^2 + 3x -6. Find each of the following. ƒ(-0.5)

Identify functions using algebra

In Exercises 11–26, determine whether each equation defines y as a function of x. y²= x

In Exercises 11–26, determine whether each equation defines y as a function of x. x + y = 16

Determine whether the three points are collinear. See Example 4. (0,-7),(-3,5),(2,-15)

Identify functions in various forms

Describe the rate of change of linear and non-linear functions

Determine whether each function is even, odd, or neither. See Example 5. ƒ(x)=0.5x^4-2x^2+6

Definition of a Function

Evaluate each function at the given values of the independent variable and simplify.  (a) f(-2), (b) f(1), (c) f(2)

Determine whether each equation defines y as a function of x. 
y = 6 -x^2

Use the graph to determine (a) the function's domain, (b) the function's range, (c) the x-intercepts, if any, (d) the y-intercept, if there is one, (e) intervals on which the function is increasing, decreasing or constant, (f) the missing function values, indicated by question marks, below each graph. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/4778f8e95e9fe698-1670872734180.jpg" alt="" />

Ex 2:  Determine a Function Value From a Graph

Determine whether each relation is a funciton, Give the domain and range for each relation. (1, 10), (2, 500), (13, π)

Determine whether each relation defines a function, and give the domain and range. See Examples 1–4.

Determine whether each function is even, odd, or neither. See Example 5. ƒ(x)=x^4-5x+8

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify.f(x)=4x+5

a. f(6)

In Exercises 39–50, graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = x², g(x) = x² - 2

Determine whether each statement is true or false. If false, explain why.
The graph of y = x^2 + 2 has no x-intercepts.

Fill in the blank(s) to correctly complete each sentence. The function g(x)=√x has domain ________.

Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. ƒ(2m-3)

For each function, find (a) ƒ(2) and (b) ƒ(-1).See Example 7. ƒ = {(-1,3),(4,7),(0,6),(2,2)}

Determine whether each function is even, odd, or neither. See Example 5. ƒ(x)=x^3-x+9

❖ The Difference Quotient - Example 1 ❖

In Exercises 55–64, use the vertical line test to identify graphs in which y is a function of x.

Use the vertical line test to identify graphs in which y is a function of x. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/a606791b04a0c430-1671046995981.jpg" alt="" />

Evaluating Piecewise Defined Functions

In Exercises 77–92, use the graph to determine a. the function’s domain; b. the function’s range; c. the x-intercepts, if any; d. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph.

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. h(x) = x³ − x + 1 

b. h (-2)

Domain and Range - Basic Idea - Two Graph Examples

Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. ƒ(1/3)

Find the coordinates of the other endpoint of each line segment, given its midpoint and one endpoint. See Example 5(b). midpoint (6a, 6b), endpoint (3a, 5b)

For the points P and Q, find (a) the distance d(P, Q) and (b) the coordinates of the mid-point M of line segment PQ. See Examples 2 and 5(a). P(-5,-6), Q(7,-1)

Finding intercepts of a nonlinear function given its graph

In Exercises 65–70, use the graph of f to find each indicated function value.  f(-3)

Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. ƒ(a+4)

In Exercises 11–26, determine whether each equation defines y as a function of x. |x| − y = 2

Determine whether the three points are the vertices of a right triangle. See Example 3. (-2,-5),(1,7),(3,15)

In Exercises 55–64, use the vertical line test to identify graphs in which y is a function of x. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/0a5bddb5a29a4972-1673544876379.jpg" alt="" />

For the points P and Q, find (a) the distance d(P, Q) and (b) the coordinates of the mid-point M of line segment PQ. See Examples 2 and 5(a). P(8,2), Q(3,5)

In Exercises 1–10, determine whether each relation is a function. Give the domain and range for each relation.{(1, 2), (3, 4), (5, 5)}

Identify polynomial extrema

In Exercises 77–92, use the graph to determine a. the function’s domain; b.the x-intercepts, if any; and e. the missing function values, indicated by question marks, below each graph. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/7e8e04a042fdfbf9-1673277839528.jpg" alt="" /><img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/d3f659d26162aa07-1678168244913.jpg" style="width: 300px;">

Find the Formula for a Piecewise Function from Graph

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify.f(x)=4x+5

c. f(-x)

In Exercises 55–64, use the vertical line test to identify graphs in which y is a function of x. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/8e766a973224250a-1673545601637.jpg" alt="" />

Determine whether the three points are collinear. See Example 4. (0,9),(-3,-7),(2,-19)

Determine whether each equation defines y as a function of x. 2x + y^2 = 6

Local and Absolute Maximum and Minimum from a Graph

In Exercises 39–50, graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x)= |x|, g(x) = |x| +1

Find the coordinates of the other endpoint of each line segment, given its midpoint and one endpoint. See Example 5(b). midpoint (5, 8), endpoint (13, 10)

The Vertical Line Test

Find the coordinates of the other endpoint of each line segment, given its midpoint and one endpoint. See Example 5(b). midpoint (-9, 8), endpoint (-16, 9)

Fill in the blank to correctly complete each sentence.
The point (4,_____ ) lies on the graph of the equation y = 3x - 6.

In Exercises 1–10, determine whether each relation is a function. Give the domain and range for each relation. {(-3, -3), (-2, −2), (−1, −1), (0, 0)}

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. f(r) = √(r + 6) +3

c. f(x-6)

Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. g(-1/4)

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. f(r) = √(r + 6) +3

b. f(10)

In Exercises 1–10, determine whether each relation is a function. Give the domain and range for each relation. {(1, 4), (1, 5), (1, 6)}

Determine whether each function is even, odd, or neither. See Example 5. ƒ(x)=x+1/x^5

Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. ƒ(3t-2)

Determine whether each relation defines a function. See Example 1.

Graph piecewise functions

Determine whether each relation defines a function, and give the domain and range. See Examples 1–4. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/66d6cd59a40eafca-1677683906402.jpg" alt="" />

An equation that defines y as a function of x is given. (b) Find ƒ(3). x-4y=8

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. f(x) = |x+3|/|x + 3|

a. f(5)

Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. g(-x)

In Exercises 55–64, use the vertical line test to identify graphs in which y is a function of x. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/39c857d1534f061a-1673621408997.jpg" alt="" />

 Determining If a Relation is Also a Function - Verbal Descriptions

In Exercises 39–50, graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = |x|, g(x) = |x| − 2

Even Odd or Neither from graph

Graphing Functions by Plotting Points - Linear 

Determine whether each relation defines a function. See Example 1. {(-3,1),(4,1),(-2,7)}

In Exercises 77–92, use the graph to determine a. the function’s domain; b. the function’s range; c. the x-intercepts, if any; d. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/a0b86496557c082b-1673544084573.jpg" alt="" />

Find the coordinates of the other endpoint of each line segment, given its midpoint and one endpoint. See Example 5(b). midpoint (12, 6), endpoint (19, 16)

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. f(x) = |x+3|/|x + 3|

b. f(-5)

The Difference Quotient - Example 2

Determining If a Relation is Also a Function - Equation Format

Fill in the blank(s) to correctly complete each sentence. For the function ƒ(x) = -4x + 2, ƒ(-2)= ______.

Determine whether the three points are the vertices of a right triangle. See Example 3. (-4,1),(1,4),(-6,-1)

Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. y=-7/(x-5)

Recognize functions

An equation that defines y as a function of x is given. (b) Find ƒ(3). y+2x^2=3-x

Find domain and range of a function using a graph

Identify a relation as a function

Graphing a Piece-Wise Defined Function - Another Example

In Exercises 55–64, use the vertical line test to identify graphs in which y is a function of x. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/16c31c486b4c56bb-1673900199541.jpg" alt="" />

In Exercises 1–10, determine whether each relation is a function. Give the domain and range for each relation. {(3, −2), (5, −2), (7, 1), (4, 9)}

Fill in the blank(s) to correctly complete each sentence. The domain of the relation { (3,5), (4, 9), (10, 13) } is _____.

Graphing in Algebra: Ordered Pairs and the Coordinate Plane

Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. g(k)

Fill in the blank to correctly complete each sentence.
The point (-1, 3) lies in quadrant ________ in the rectangular coordinate system.

Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. g(1/2)

Testing an Equation for Symmetry

Fill in the blank to correctly complete each sentence.
The x-intercept of the graph of 2x + 5y = 10 is ________.

Describe function relationships using graphs

Determine whether each equation defines y as a function of x. 2x + y = 8

Graphing Equations by Plotting Points - Example 1

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. h(x) = x^4 - x²+1

c. h (-x)

In Exercises 77–92, use the graph to determine a. the function’s domain; b. the function’s range; c. the x-intercepts, if any; d. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/1a9689c3e3b8989a-1676804768575.jpg" alt="" />

Domain and range from the graph of a discrete relation

In Exercises 55–64, use the vertical line test to identify graphs in which y is a function of x. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/d997c8b190e5b9a1-1673550287859.jpg" alt="" />

Adding and Subtracting Functions - Function Notation

In Exercises 77–92, use the graph to determine a.the x-intercepts, if any; b. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/f17038ff2bb0aed2-1673459145181.jpg" alt="" />

Evaluate a function given an input value

Finding Domain and Range of a Function using a Graph

Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. ƒ(0)

Determine whether each equation defines y as a function of x. 
y = ±√(x-2)

In Exercises 77–92, use the graph to determine a.the x-intercepts, if any; b. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/c987be154430f677-1673458883182.jpg" alt="" />

Graphing Equations by Plotting Points - Example 3

Identify functions given sets, mappings, and real world scenarios

Graph a Line by Plotting Points - Example 2

Graph piecewise-defined functions

In Exercises 11–26, determine whether each equation defines y as a function of x. y = - √x +4

Determine whether the three points are the vertices of a right triangle. See Example 3. (-4,3),(2,5),(-1,-6)

Find the value of the function for the given value of x. See Example 3. ƒ(x)=-[[-x]], for x=2.5

In Exercises 55–64, use the vertical line test to identify graphs in which y is a function of x. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/3a22f94aa6b6db4a-1673549344708.jpg" alt="" />

Determine whether each relation defines a function, and give the domain and range. See Examples 1–4. {(1,1),(1,-1),(0,0),(2,4),(2,-4)}

Find intervals that a function is increasing and decreasing

ACT PREP - Find the Domain of a Function

 Using a Graph to Determine Values of a Function

How to calculate the difference quotient

Determine whether each equation defines y as a function of x. 
x = (1/3)(y^2)

In Exercises 95–96, let f and g be defined by the following table: Find |ƒ(1) − f(0)| − [g (1)]² +g(1) ÷ ƒ(−1) · g (2) .

Determine whether the three points are the vertices of a right triangle. See Example 3. (-2,-8),(0,-4),(-4,-7)

Determine whether the three points are the vertices of a right triangle. See Example 3. (-6,-4),(0,-2),(-10,8)

In Exercises 51–54, graph the given square root functions, f and g, in the same rectangular coordinate system. Use the integer values of x given to the right of each function to obtain ordered pairs. Because only nonnegative numbers have square roots that are real numbers, be sure that each graph appears only for values of x that cause the expression under the radical sign to be greater than or equal to zero. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = √x (x = 0, 1, 4, 9) and g(x) = √x −1 (x = 0, 1, 4, 9)

Determine whether each equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these. 
5y^2 + 5x^2 =30

Use the graph of g to solve Exercises 71–76. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/bc88945b53f967a6-1673552008339.jpg" alt="" /> Find g(-4)

Recognize functions using the vertical line test

What Is Function Notation?

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. h(x) = x³ − x + 1 

d. h (3a)

Fill in the blank(s) to correctly complete each sentence. The equation y = 4x - 6 defines a function with independent variable______ 
and dependent variable ________ .

Finding Intercepts from the Graph of Functions

Determine whether each relation defines a function, and give the domain and range. See Examples 1–4. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/d03c079ba4193d7a-1677682584282.jpg" alt="" />

In Exercises 11–26, determine whether each equation defines y as a function of x. x+y³ = 8

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. g(x) = x² - 10x - 3

b. g(x+2)In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify.g(x) = x² + 2x + 3

b. g(x+5)

Determine whether each relation defines a function, and give the domain and range. See Examples 1–4. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/beb9adf6d6bedf03-1677680955762.jpg" alt="" />

In Exercises 1–10, determine whether each relation is a function. Give the domain and range for each relation. {(3, 4), (3, 5), (4, 4), (4, 5)}

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify.g(x) = x² + 2x + 3

a. g(-1)

The Difference Quotient - Example 1

For each piecewise-defined function, find (a) ƒ(-5), (b) ƒ(-1), (c) ƒ(0), and (d) ƒ(3).See Example 2. ƒ(x)={2+x if x<-4, -x if -4≤x≤2, 3x if x>2

Evaluate each function at the given values of the independent variable and simplify. g(x) = 3x^2 - 5x + 2 (a) g(0), (b) g(-2), (c) g(x-1), (d) g(-x)

Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. ƒ(-3)

In Exercises 93-94, let f(x) = x² − x + 4 and g(x) = 3x – 5. Find g(-1) and f(g(-1)).

Determine whether each relation defines a function, and give the domain and range. See Examples 1–4. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/0ce019cdb0798d0b-1677681271682.jpg" alt="" />

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. h(x) = x³ − x + 1 

a. h (3)

Graphing functions by plotting points

Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. ƒ(-x)

Use the graph of g to solve Exercises 71–76. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/ea7eeab6f5daef5a-1674737508484.jpg" alt="" /> For what value of x is g(x) = 1?

Even/Odd Functions

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. f(r) = √(r + 6) +3

a. f(-6)

Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. ƒ(x+2)

Determining If a Relation is Also a Function - Ordered Pairs1

Find the value of the function for the given value of x. See Example 3. ƒ(x)=2-[[-x]], for x=3.7

Distinguish between even and odd functions

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. h(x) = x^4 - x²+1

a. h (2)

Determine whether each statement is true or false. If false, explain why.
The midpoint of the segment joining (0, 0) and (4, 4) is 2.

In Exercises 51–54, graph the given square root functions, f and g, in the same rectangular coordinate system. Use the integer values of x given to the right of each function to obtain ordered pairs. Because only nonnegative numbers have square roots that are real numbers, be sure that each graph appears only for values of x that cause the expression under the radical sign to be greater than or equal to zero. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = √x (x = 0, 1, 4, 9) and g (x) = √(x + 2) (x = = −2, −1, 2, 7)

Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. g(10)

Determining If a Relation is Also a Function

How to determine the intervals that a function is increasing decreasing or constant

In Exercises 55–64, use the vertical line test to identify graphs in which y is a function of x. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/2aae7522dd49b455-1673625604921.jpg" alt="" />

In Exercises 11–26, determine whether each equation defines y as a function of x. y = √x +4

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. f(x) = |x+3|/|x + 3|

c. f(−9 - x)

In Exercises 11–26, determine whether each equation defines y as a function of x. xy - 5y =1

In Exercises 65–70, use the graph of f to find each indicated function value.  f(4)

Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. g(-2)

In Exercises 65–70, use the graph of f to find each indicated function value. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/bc6a92a3e935b5ef-1676041601888.jpg" alt="" /> f(-2)

Determine the domain and range of a parabola: looking at the graph

Determine whether each relation defines a function. See Example 1. {(5,1),(3,2),(4,9),(7,8)}

Testing an equation for symmetry about the axes and origin

In Exercises 39–50, graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = x, g(x) = x + 3

Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. ƒ(-7/3)

Determine whether each relation defines a function. See Example 1. {(8,0),(5,7),(9,3),(3,8)}

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. g(x) = x² - 10x - 3

c. g(-x)In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify.g(x) = x² + 2x + 3

c. g(-x)

Determine whether each function is even, odd, or neither. See Example 5. ƒ(x)=x^4+4/x^2

In Exercises 77–92, use the graph to determine a. the function’s range; b. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/60b1f9575e169ea3-1673276595459.jpg" alt="" />

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. h(x) = x³ − x + 1 

c. h (-x)

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. h(x) = x^4 - x²+1

d. h (3a)

In Exercises 77–92, use the graph to determine a.the x-intercepts, if any; b. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/824c1dfe8e7e137b-1673459239394.jpg" alt="" />

Simplify: 2(x+h)² + 3 (x + h) + 5 − (2x² + 3x + 5).

Determine whether the graph of each equation is symmetric with respect to the y-axis, the x-axis, the origin, more than one of those, or none of these. x^2 + y^2 =17

In Exercises 93-94, let f(x) = x² − x + 4 and g(x) = 3x – 5. Find g (1) and f(g(1)).