Learn on PengiOpenstax Intermediate Algebra 2EChapter 3: Graphs and Functions

Lesson 3.6: Graphs of Functions

New Concept This lesson introduces visual tools for understanding functions. You'll learn to use the vertical line test to confirm if a graph is a function, recognize common function graphs, and read key information like domain and range directly from them.

Section 1

📘 Graphs of Functions

New Concept

This lesson introduces visual tools for understanding functions. You'll learn to use the vertical line test to confirm if a graph is a function, recognize common function graphs, and read key information like domain and range directly from them.

What’s next

First, you'll master the vertical line test with interactive examples. Then, you'll explore the graphs of basic functions through a series of practice cards.

Section 2

Vertical Line Test

Property

A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point.

If any vertical line intersects the graph in more than one point, the graph does not represent a function.

Examples

  • The graph of a parabola opening upwards, like y=x2+1y = x^2 + 1, is a function because any vertical line intersects it only once.

Section 3

Graph of a Function

Property

The graph of a function is the graph of all its ordered pairs, (x,y)(x, y) or using function notation, (x,f(x))(x, f(x)) where y=f(x)y = f(x).

ffname of function
xxxx-coordinate of the ordered pair
f(x)f(x)yy-coordinate of the ordered pair

Examples

  • For the function f(x)=2x+1f(x) = 2x + 1, the point (3,7)(3, 7) is on its graph because f(3)=2(3)+1=7f(3) = 2(3) + 1 = 7.

Section 4

Linear, Constant, and Identity Functions

Property

FunctionDefinitionDomainRange
Linear Functionf(x)=mx+bf(x) = mx + b(−∞,∞)(-\infty, \infty)(−∞,∞)(-\infty, \infty)
Constant Functionf(x)=bf(x) = b(−∞,∞)(-\infty, \infty){b}\{b\}
Identity Functionf(x)=xf(x) = x(−∞,∞)(-\infty, \infty)(−∞,∞)(-\infty, \infty)

Examples

  • The linear function f(x)=3x−2f(x) = 3x - 2 creates a line with a slope of 3 and a y-intercept at (0,−2)(0, -2).

Section 5

Square and Cube Functions

Property

FunctionDefinitionDomainRange
Square Functionf(x)=x2f(x) = x^2(−∞,∞)(-\infty, \infty)[0,∞)[0, \infty)
Cube Functionf(x)=x3f(x) = x^3(−∞,∞)(-\infty, \infty)(−∞,∞)(-\infty, \infty)

Examples

  • For the square function f(x)=x2f(x) = x^2, both x=4x=4 and x=−4x=-4 produce the same output, f(x)=16f(x)=16, showing its symmetry.

Section 6

Square Root and Absolute Value Functions

Property

FunctionDefinitionDomainRange
Square Root Functionf(x)=xf(x) = \sqrt{x}[0,∞)[0, \infty)[0,∞)[0, \infty)
Absolute Value Functionf(x)=∣x∣f(x) = \lvert x \rvert(−∞,∞)(-\infty, \infty)[0,∞)[0, \infty)

Examples

  • The function f(x)=xf(x) = \sqrt{x} is undefined for negative inputs, so its domain starts at 0. For example, f(25)=5f(25) = 5.

Section 7

Reading Information from a Graph

Property

Domain: The set of all the xx-values in the ordered pairs in the function. To find the domain we look at the graph and find all the values of xx that have a corresponding value on the graph.

Range: The set of all the yy-values in the ordered pairs in the function. To find the range we look at the graph and find all the values of yy that have a corresponding value on the graph.

Examples

  • If a graph extends horizontally from x=−5x=-5 to x=5x=5, its domain is [−5,5][-5, 5]. If it extends vertically from y=0y=0 to y=10y=10, its range is [0,10][0, 10].

Book overview

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Chapter 3: Graphs and Functions

  1. Lesson 1

    Lesson 3.1: Graph Linear Equations in Two Variables

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  2. Lesson 2

    Lesson 3.2: Slope of a Line

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  3. Lesson 3

    Lesson 3.3: Find the Equation of a Line

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  4. Lesson 4

    Lesson 3.4: Graph Linear Inequalities in Two Variables

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  5. Lesson 5

    Lesson 3.5: Relations and Functions

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  6. Lesson 6Current

    Lesson 3.6: Graphs of Functions

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Lesson overview

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Section 1

📘 Graphs of Functions

New Concept

This lesson introduces visual tools for understanding functions. You'll learn to use the vertical line test to confirm if a graph is a function, recognize common function graphs, and read key information like domain and range directly from them.

What’s next

First, you'll master the vertical line test with interactive examples. Then, you'll explore the graphs of basic functions through a series of practice cards.

Section 2

Vertical Line Test

Property

A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point.

If any vertical line intersects the graph in more than one point, the graph does not represent a function.

Examples

  • The graph of a parabola opening upwards, like y=x2+1y = x^2 + 1, is a function because any vertical line intersects it only once.

Section 3

Graph of a Function

Property

The graph of a function is the graph of all its ordered pairs, (x,y)(x, y) or using function notation, (x,f(x))(x, f(x)) where y=f(x)y = f(x).

ffname of function
xxxx-coordinate of the ordered pair
f(x)f(x)yy-coordinate of the ordered pair

Examples

  • For the function f(x)=2x+1f(x) = 2x + 1, the point (3,7)(3, 7) is on its graph because f(3)=2(3)+1=7f(3) = 2(3) + 1 = 7.

Section 4

Linear, Constant, and Identity Functions

Property

FunctionDefinitionDomainRange
Linear Functionf(x)=mx+bf(x) = mx + b(−∞,∞)(-\infty, \infty)(−∞,∞)(-\infty, \infty)
Constant Functionf(x)=bf(x) = b(−∞,∞)(-\infty, \infty){b}\{b\}
Identity Functionf(x)=xf(x) = x(−∞,∞)(-\infty, \infty)(−∞,∞)(-\infty, \infty)

Examples

  • The linear function f(x)=3x−2f(x) = 3x - 2 creates a line with a slope of 3 and a y-intercept at (0,−2)(0, -2).

Section 5

Square and Cube Functions

Property

FunctionDefinitionDomainRange
Square Functionf(x)=x2f(x) = x^2(−∞,∞)(-\infty, \infty)[0,∞)[0, \infty)
Cube Functionf(x)=x3f(x) = x^3(−∞,∞)(-\infty, \infty)(−∞,∞)(-\infty, \infty)

Examples

  • For the square function f(x)=x2f(x) = x^2, both x=4x=4 and x=−4x=-4 produce the same output, f(x)=16f(x)=16, showing its symmetry.

Section 6

Square Root and Absolute Value Functions

Property

FunctionDefinitionDomainRange
Square Root Functionf(x)=xf(x) = \sqrt{x}[0,∞)[0, \infty)[0,∞)[0, \infty)
Absolute Value Functionf(x)=∣x∣f(x) = \lvert x \rvert(−∞,∞)(-\infty, \infty)[0,∞)[0, \infty)

Examples

  • The function f(x)=xf(x) = \sqrt{x} is undefined for negative inputs, so its domain starts at 0. For example, f(25)=5f(25) = 5.

Section 7

Reading Information from a Graph

Property

Domain: The set of all the xx-values in the ordered pairs in the function. To find the domain we look at the graph and find all the values of xx that have a corresponding value on the graph.

Range: The set of all the yy-values in the ordered pairs in the function. To find the range we look at the graph and find all the values of yy that have a corresponding value on the graph.

Examples

  • If a graph extends horizontally from x=−5x=-5 to x=5x=5, its domain is [−5,5][-5, 5]. If it extends vertically from y=0y=0 to y=10y=10, its range is [0,10][0, 10].

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 3: Graphs and Functions

  1. Lesson 1

    Lesson 3.1: Graph Linear Equations in Two Variables

    LearnPractice
  2. Lesson 2

    Lesson 3.2: Slope of a Line

    LearnPractice
  3. Lesson 3

    Lesson 3.3: Find the Equation of a Line

    LearnPractice
  4. Lesson 4

    Lesson 3.4: Graph Linear Inequalities in Two Variables

    LearnPractice
  5. Lesson 5

    Lesson 3.5: Relations and Functions

    LearnPractice
  6. Lesson 6Current

    Lesson 3.6: Graphs of Functions

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