practice assignment 3.1 functions and function notation

MAT 105: College Algebra: 3.1 Functions and Function Notation

• Getting Started
• 1.1 Real Numbers: Algebra Essentials
• 1.2 Exponents and Scientific Notation
• 1.3 Radicals and Rational Exponents
• 1.4 Polynomials
• 1.5 Factoring Polynomials
• 1.6 Rational Expressions
• 2.1 The Rectangular Coordinate Systems and Graphs
• 2.2 Linear Equations in One Variable
• 2.3 Models and Applications
• 2.4 Complex Numbers
• 2.5 Quadratic Equations
• 2.6 Other Types of Equations
• 2.7 Linear Inequalities and Absolute Value Inequalities

3.1 Functions and Function Notation

• 3.2 Domain and Range
• 3.3 Rates of Change and Behavior of Graphs
• 3.4 Composition of Functions
• 3.5 Transformation of Functions
• 3.6 Absolute Value Functions
• 3.7 Inverse Functions
• 4.1 Linear Functions
• 4.2 Modeling with Linear Functions
• 4.3 Fitting Linear Models to Data
• 5.1 Quadratic Functions
• 5.2 Power Functions and Polynomial Functions
• 5.3 Graphs of Polynomial Functions
• 5.4 Dividing Polynomials
• 5.5 Zeros of Polynomial Functions
• 5.6 Rational Functions
• 5.7 Inverses and Radical Functions
• 5.8 Modeling Using Variation
• 6.1 Exponential Functions
• 6.2 Graphs of Exponential Functions
• 6.3 Logarithmic Functions
• 6.4 Graphs of Logarithmic Functions
• 6.5 Logarithmic Properties
• 6.6 Exponential and Logarithmic Equations
• 6.7 Exponential and Logarithmic Models
• 6.8 Fitting Exponential Models to Data
• 7.1 Systems of Linear Equations: Two Variables
• 7.2 Systems of Linear Equations: Three Variables
• 7.3 Systems of Nonlinear Equations and Inequalities: Two Variables

Learning Objectives

In this section, you will:

• Determine whether a relation represents a function,
• Find the value of a function,
• Determine whether a function is a one-to-one,
• Use the vertical line test to identify functions,
• Graph the functions listed in the library of functions.

Determine Whether a Relation Represents a Function

End of Slides for This Section

Find the Value of a Function

Determine Whether a Function is One-to-One

Use the Vertical Line Test to Identify Functions

Graph the Functions Listed in the Library of Functions

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• Next: 3.2 Domain and Range >>
• Last Updated: May 2, 2024 11:23 AM
• URL: https://garrettcollege.libguides.com/c.php?g=1388750

Introduction to Functions

Chapter outline.

Toward the end of the twentieth century, the values of stocks of Internet and technology companies rose dramatically. As a result, the Standard and Poor’s stock market average rose as well. The graph above tracks the value of that initial investment of just under $100 over the 40 years. It shows that an investment that was worth less than $500 until about 1995 skyrocketed up to about $1100 by the beginning of 2000. That five-year period became known as the “dot-com bubble” because so many Internet startups were formed. As bubbles tend to do, though, the dot-com bubble eventually burst. Many companies grew too fast and then suddenly went out of business. The result caused the sharp decline represented on the graph beginning at the end of 2000.

Notice, as we consider this example, that there is a definite relationship between the year and stock market average. For any year we choose, we can determine the corresponding value of the stock market average. In this chapter, we will explore these kinds of relationships and their properties.

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Access for free at https://openstax.org/books/college-algebra-2e/pages/1-introduction-to-prerequisites

• Authors: Jay Abramson
• Publisher/website: OpenStax
• Book title: College Algebra 2e
• Publication date: Dec 21, 2021
• Location: Houston, Texas
• Book URL: https://openstax.org/books/college-algebra-2e/pages/1-introduction-to-prerequisites
• Section URL: https://openstax.org/books/college-algebra-2e/pages/3-introduction-to-functions

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College Algebra

R. david gustafson, jeffrey d. hughes, functions and function notation - all with video answers.

Functions and Function Notation

You should be able to complete these vocabulary and concept statements before you proceed to the practice exercises. Fill in the blanks. A correspondence that assigns exactly one value of y to any number x is called a ______.

You should be able to complete these vocabulary and concept statements before you proceed to the practice exercises. Fill in the blanks. A correspondence that assigns one or more values of $y$ to any number $x$ is called a _____.

You should be able to complete these vocabulary and concept statements before you proceed to the practice exercises. Fill in the blanks. The set of input numbers $x$ in a function is called the _____ of the function.

You should be able to complete these vocabulary and concept statements before you proceed to the practice exercises. Fill in the blanks. The set of all output values $y$ in a function is called the ______ of the function.

You should be able to complete these vocabulary and concept statements before you proceed to the practice exercises. Fill in the blanks. The statement " $y$ is a function of $x$ " can be written as the equation _______.

You should be able to complete these vocabulary and concept statements before you proceed to the practice exercises. Fill in the blanks. The graph of a function $y=f(x)$ in the $x y$ -plane is the set of all points ______ that satisfy the equation, where $x$ is in the ______ of $f$ and $y$ is in the _____ of $f$

You should be able to complete these vocabulary and concept statements before you proceed to the practice exercises. Fill in the blanks. In the function of Exercise $5, ______ $ is called the independent variable.

You should be able to complete these vocabulary and concept statements before you proceed to the practice exercises. Fill in the blanks. In the function of Exercise $5, y$ is called the ______ variable.

You should be able to complete these vocabulary and concept statements before you proceed to the practice exercises. Fill in the blanks. If every ______ line that intersects a graph does so ______ , the graph represents a function.

You should be able to complete these vocabulary and concept statements before you proceed to the practice exercises. Fill in the blanks. A function that can be written in the form $y=m x+b$ is called a ______ function.

Assume that all variables represent real numbers. Determine whether each equation determines $y$ to be a function of $x$. $$y=x$$

Assume that all variables represent real numbers. Determine whether each equation determines $y$ to be a function of $x$. $$y-2 x=0$$

Assume that all variables represent real numbers. Determine whether each equation determines $y$ to be a function of $x$. $$y^{2}=x$$

Assume that all variables represent real numbers. Determine whether each equation determines $y$ to be a function of $x$. $$|y|=x$$

Assume that all variables represent real numbers. Determine whether each equation determines $y$ to be a function of $x$.

Assume that all variables represent real numbers. Determine whether each equation determines $y$ to be a function of $x$. $$y-7=7$$

Assume that all variables represent real numbers. Determine whether each equation determines $y$ to be a function of $x$. $$y^{2}-4 x=1$$

Assume that all variables represent real numbers. Determine whether each equation determines $y$ to be a function of $x$. $$|x-2|=y$$

Assume that all variables represent real numbers. Determine whether each equation determines $y$ to be a function of $x$. $$|x|=|y|$$

Assume that all variables represent real numbers. Determine whether each equation determines $y$ to be a function of $x$. $$x=7$$

Assume that all variables represent real numbers. Determine whether each equation determines $y$ to be a function of $x$. $$y=7$$

Assume that all variables represent real numbers. Determine whether each equation determines $y$ to be a function of $x$. $$|x+y|=7$$

Let the function $f$ be defined by the equation $y=f(x)$ where $x$ and $f(x)$ are real numbers. Find the domain of each function. $$f(x)=3 x+5$$

Let the function $f$ be defined by the equation $y=f(x)$ where $x$ and $f(x)$ are real numbers. Find the domain of each function. $$f(x)=-5 x+2$$

Let the function $f$ be defined by the equation $y=f(x)$ where $x$ and $f(x)$ are real numbers. Find the domain of each function. $$f(x)=x^{2}-x+1$$

Let the function $f$ be defined by the equation $y=f(x)$ where $x$ and $f(x)$ are real numbers. Find the domain of each function. $$f(x)=x^{3}-3 x+2$$

Let the function $f$ be defined by the equation $y=f(x)$ where $x$ and $f(x)$ are real numbers. Find the domain of each function. $$f(x)=\sqrt{x-2}$$

Let the function $f$ be defined by the equation $y=f(x)$ where $x$ and $f(x)$ are real numbers. Find the domain of each function. $$f(x)=\sqrt{2 x+3}$$

Let the function $f$ be defined by the equation $y=f(x)$ where $x$ and $f(x)$ are real numbers. Find the domain of each function. $$f(x)=\sqrt{4-x}$$

Let the function $f$ be defined by the equation $y=f(x)$ where $x$ and $f(x)$ are real numbers. Find the domain of each function. $$f(x)=3 \sqrt{2-x}$$

Let the function $f$ be defined by the equation $y=f(x)$ where $x$ and $f(x)$ are real numbers. Find the domain of each function. $$f(x)=\sqrt{x^{2}-1}$$

Let the function $f$ be defined by the equation $y=f(x)$ where $x$ and $f(x)$ are real numbers. Find the domain of each function. $$f(x)=\sqrt{x^{2}-2 x-3}$$

Let the function $f$ be defined by the equation $y=f(x)$ where $x$ and $f(x)$ are real numbers. Find the domain of each function. $$f(x)=\sqrt[3]{x+1}$$

Let the function $f$ be defined by the equation $y=f(x)$ where $x$ and $f(x)$ are real numbers. Find the domain of each function. $$f(x)=\sqrt[3]{5-x}$$

Let the function $f$ be defined by the equation $y=f(x)$ where $x$ and $f(x)$ are real numbers. Find the domain of each function. $$f(x)=\frac{3}{x+1}$$

Let the function $f$ be defined by the equation $y=f(x)$ where $x$ and $f(x)$ are real numbers. Find the domain of each function. $$f(x)=\frac{-7}{x+3}$$

Let the function $f$ be defined by the equation $y=f(x)$ where $x$ and $f(x)$ are real numbers. Find the domain of each function. $$f(x)=\frac{x}{x-3}$$

Let the function $f$ be defined by the equation $y=f(x)$ where $x$ and $f(x)$ are real numbers. Find the domain of each function. $$f(x)=\frac{x+2}{x-1}$$

Let the function $f$ be defined by the equation $y=f(x)$ where $x$ and $f(x)$ are real numbers. Find the domain of each function. $$f(x)=\frac{x}{x^{2}-4}$$

Let the function $f$ be defined by the equation $y=f(x)$ where $x$ and $f(x)$ are real numbers. Find the domain of each function. $$f(x)=\frac{2 x}{x^{2}-9}$$

Let the function $f$ be defined by the equation $y=f(x)$ where $x$ and $f(x)$ are real numbers. Find the domain of each function. $$f(x)=\frac{1}{x^{2}-4 x-5}$$

Let the function $f$ be defined by the equation $y=f(x)$ where $x$ and $f(x)$ are real numbers. Find the domain of each function. $$f(x)=\frac{x}{2 x^{2}-16 x+30}$$

Let the function $f$ be defined by $y=f(x),$ where $x$ and $f(x)$ are real mumbers. Find $f(\mathbf{2}), f(-3), f(k),$ and $f\left(k^{2}-1\right)$ $$f(x)=3 x-2$$

Let the function $f$ be defined by $y=f(x),$ where $x$ and $f(x)$ are real mumbers. Find $f(\mathbf{2}), f(-3), f(k),$ and $f\left(k^{2}-1\right)$ $$f(x)=5 x+7$$

Let the function $f$ be defined by $y=f(x),$ where $x$ and $f(x)$ are real mumbers. Find $f(\mathbf{2}), f(-3), f(k),$ and $f\left(k^{2}-1\right)$ $$f(x)=\frac{1}{2} x+3$$

Let the function $f$ be defined by $y=f(x),$ where $x$ and $f(x)$ are real mumbers. Find $f(\mathbf{2}), f(-3), f(k),$ and $f\left(k^{2}-1\right)$ $$f(x)=\frac{2}{3} x+5$$

Let the function $f$ be defined by $y=f(x),$ where $x$ and $f(x)$ are real mumbers. Find $f(\mathbf{2}), f(-3), f(k),$ and $f\left(k^{2}-1\right)$ $$f(x)=x^{2}$$

Let the function $f$ be defined by $y=f(x),$ where $x$ and $f(x)$ are real mumbers. Find $f(\mathbf{2}), f(-3), f(k),$ and $f\left(k^{2}-1\right)$ $$f(x)=3-x^{2}$$

Let the function $f$ be defined by $y=f(x),$ where $x$ and $f(x)$ are real mumbers. Find $f(\mathbf{2}), f(-3), f(k),$ and $f\left(k^{2}-1\right)$ $$f(x)=x^{2}+3 x-1$$

Let the function $f$ be defined by $y=f(x),$ where $x$ and $f(x)$ are real mumbers. Find $f(\mathbf{2}), f(-3), f(k),$ and $f\left(k^{2}-1\right)$ $$f(x)=-x^{2}-2 x+1$$

Let the function $f$ be defined by $y=f(x),$ where $x$ and $f(x)$ are real mumbers. Find $f(\mathbf{2}), f(-3), f(k),$ and $f\left(k^{2}-1\right)$ $$f(x)=\left|x^{2}+1\right|$$

Let the function $f$ be defined by $y=f(x),$ where $x$ and $f(x)$ are real mumbers. Find $f(\mathbf{2}), f(-3), f(k),$ and $f\left(k^{2}-1\right)$ $$f(x)=\left|x^{2}+x+4\right|$$

Let the function $f$ be defined by $y=f(x),$ where $x$ and $f(x)$ are real mumbers. Find $f(\mathbf{2}), f(-3), f(k),$ and $f\left(k^{2}-1\right)$ $$f(x)=\frac{2}{x+4}$$

Let the function $f$ be defined by $y=f(x),$ where $x$ and $f(x)$ are real mumbers. Find $f(\mathbf{2}), f(-3), f(k),$ and $f\left(k^{2}-1\right)$ $$f(x)=\frac{3}{x-5}$$

Let the function $f$ be defined by $y=f(x),$ where $x$ and $f(x)$ are real mumbers. Find $f(\mathbf{2}), f(-3), f(k),$ and $f\left(k^{2}-1\right)$ $$f(x)=\frac{1}{x^{2}-1}$$

Let the function $f$ be defined by $y=f(x),$ where $x$ and $f(x)$ are real mumbers. Find $f(\mathbf{2}), f(-3), f(k),$ and $f\left(k^{2}-1\right)$ $$f(x)=\frac{3}{x^{2}+3}$$

Let the function $f$ be defined by $y=f(x),$ where $x$ and $f(x)$ are real mumbers. Find $f(\mathbf{2}), f(-3), f(k),$ and $f\left(k^{2}-1\right)$ $$f(x)=\sqrt{x^{2}+1}$$

Let the function $f$ be defined by $y=f(x),$ where $x$ and $f(x)$ are real mumbers. Find $f(\mathbf{2}), f(-3), f(k),$ and $f\left(k^{2}-1\right)$ $$f(x)=\sqrt{x^{2}-1}$$

Evaluate the difference quotient for each function $f(x)$ $$f(x)=3 x+1$$

Evaluate the difference quotient for each function $f(x)$ $$f(x)=5 x-1$$

Evaluate the difference quotient for each function $f(x)$ $$f(x)=x^{2}+1$$

Evaluate the difference quotient for each function $f(x)$ $$f(x)=x^{2}-3$$

Evaluate the difference quotient for each function $f(x)$ $$f(x)=4 x^{2}-6$$

Evaluate the difference quotient for each function $f(x)$ $$f(x)=5 x^{2}+3$$

Evaluate the difference quotient for each function $f(x)$ $$f(x)=x^{2}+3 x-7$$

Evaluate the difference quotient for each function $f(x)$ $$f(x)=x^{2}-5 x+1$$

Evaluate the difference quotient for each function $f(x)$ $$f(x)=2 x^{2}-4 x+2$$

Evaluate the difference quotient for each function $f(x)$ $$f(x)=3 x^{2}+2 x-3$$

Evaluate the difference quotient for each function $f(x)$ $$f(x)=x^{3}$$

Evaluate the difference quotient for each function $f(x)$ $$f(x)=\frac{1}{x}$$

Graph each function. Use the graph to identify the domain and range of each function. (GRAPH CANNOT COPY) $$f(x)=2 x+3$$

Graph each function. Use the graph to identify the domain and range of each function. (GRAPH CANNOT COPY) $$f(x)=3 x+2$$

Graph each function. Use the graph to identify the domain and range of each function. (GRAPH CANNOT COPY) $$f(x)=-\frac{3}{4} x+4$$

Graph each function. Use the graph to identify the domain and range of each function. (GRAPH CANNOT COPY) $$f(x)=\frac{1}{2} x-3$$

Graph each function. Use the graph to identify the domain and range of each function. (GRAPH CANNOT COPY) $$2 x=3 y-3$$

Graph each function. Use the graph to identify the domain and range of each function. (GRAPH CANNOT COPY) $$3 x=2(y+1)$$

Graph each function. Use the graph to identify the domain and range of each function. (GRAPH CANNOT COPY) $$f(x)=x^{2}-4$$

Graph each function. Use the graph to identify the domain and range of each function. (GRAPH CANNOT COPY) $$f(x)=-x^{2}+3$$

Graph each function. Use the graph to identify the domain and range of each function. (GRAPH CANNOT COPY) $$f(x)=-x^{3}+2$$

Graph each function. Use the graph to identify the domain and range of each function. (GRAPH CANNOT COPY) $$f(x)=-x^{3}+1$$

Graph each function. Use the graph to identify the domain and range of each function. (GRAPH CANNOT COPY) $$f(x)=-|x|$$

Graph each function. Use the graph to identify the domain and range of each function. (GRAPH CANNOT COPY) $$f(x)=-|x|-3$$

Graph each function. Use the graph to identify the domain and range of each function. (GRAPH CANNOT COPY) $$f(x)=|x-2|$$

Graph each function. Use the graph to identify the domain and range of each function. (GRAPH CANNOT COPY) $$f(x)=-|x-2|$$

Graph each function. Use the graph to identify the domain and range of each function. (GRAPH CANNOT COPY) $$f(x)=\left|\frac{1}{2} x+3\right|$$

Graph each function. Use the graph to identify the domain and range of each function. (GRAPH CANNOT COPY) $$f(x)=-\left|\frac{1}{2} x+3\right|$$

Graph each function. Use the graph to identify the domain and range of each function. (GRAPH CANNOT COPY) $$f(x)=-\sqrt{x+1}$$

Graph each function. Use the graph to identify the domain and range of each function. (GRAPH CANNOT COPY) $$f(x)=\sqrt{x}+2$$

Graph each function. Use the graph to identify the domain and range of each function. (GRAPH CANNOT COPY) $$f(x)=\sqrt{2 x-4}$$

Graph each function. Use the graph to identify the domain and range of each function. (GRAPH CANNOT COPY) $$f(x)=-\sqrt{2 x-4}$$

Graph each function. Use the graph to identify the domain and range of each function. (GRAPH CANNOT COPY) $$f(x)=\sqrt[3]{x}+2$$

Graph each function. Use the graph to identify the domain and range of each function. (GRAPH CANNOT COPY) $$f(x)=-\sqrt[3]{x}+1$$

Draw lines to indicate the domain and range of each function as intervals on the $x$ - and $y$ -axes. (GRAPH CANNOT COPY)

Use the Vertical Line Test to determine whether each graph represents a function. (GRAPH CANNOT COPY)

Problem 100

Problem 101.

Use a graphing calculator to graph each function. Then determine the domain and range of the function. $$f(x)=|3 x+2|$$

Problem 102

Use a graphing calculator to graph each function. Then determine the domain and range of the function. $$f(x)=\sqrt{2 x-5}$$

Problem 103

Use a graphing calculator to graph each function. Then determine the domain and range of the function. $$f(x)=\sqrt[3]{5 x-1}$$

Problem 104

Use a graphing calculator to graph each function. Then determine the domain and range of the function. $$f(x)=-\sqrt[3]{3 x+2}$$

Problem 105

cost of $t$ -shirts A chapter of Phi Theta Kappa, an honors society for two-year college students, is purchasing $t$ -shirts for each of its members. A local company has agreed to make the shirts for 8 dollars each plus a graphic arts fee of 75 dollars . a. Write a linear function that describes the cost C for the shirts in terms of $x,$ the number of $t$ -shirts ordered. b. Find the total cost of $85 ~ t-$ shirts.

Problem 106

Service projects The Circle "K" Club is planning a service project for children at a local children's home. They plan to rent a "Dora the Explorer Moonwalk" for the event. The cost of the moonwalk will include a 60 dollars delivery fee and 45 dollars for each hour it is used. Express the total bill $b$ in terms of the hours used $h$

Problem 107

Cell phone plans A grandmother agrees to purchase a cell phone for emergency use only. AT\&T now offers such a plan for 9.99 dollars per month and 0.07 dollars for each minute $t$ the phone is used. a. Write a linear function that describes the monthly cost $C$ in terms of the time in minutes $t$ the phone is used. b. If the grandmother uses her phone for 20 minutes during the first month, what was her bill?

Problem 108

Concessions A concessionaire at a football game pays a vendor 40 dollars per game for selling hot dogs at 2.50 dollars each. a. Write a linear function that describes the income $I$ the vendor earns for the concessionaire during the game if the vendor sells $h$ hot dogs. b. Find the income if the vendor sells 175 hot dogs.

Problem 109

Home construction In a proposal to prospective clients, a contractor listed the following costs:\ 1. Fees, permits, site preparation 14,000 dollars 2. Construction, per square foot 95 dollars a. Write a linear function the clients can use to determine the cost $C$ of building a house having $f$ square feet. b. Find the cost to build a $2,600$ -square-foot house.

Problem 110

Temperature conversion The Fahrenheit temperature reading ( $F$ ) is a linear function of the Celsius reading (C). If $C=0$ when $F=32$ and the readings are the same at $-40^{\circ},$ express $F$ as a function of $C .$

Problem 111

cost of electricity The cost $C$ of electricity in Eagle River is a linear function of $x,$ the number of kilowatt-hours (kwh) used. If the cost of $100 \mathrm{kwh}$ is 17 dollars and the cost of $500 \mathrm{kwh}$ is 57 dollars find an equation that expresses $C$ in terms of $x .$

Problem 112

Water billing The cost $C$ of water is a linear function of $n,$ the number of gallons used. If $1,000$ gallons cost 4.70 dollars and $9,000$ gallons cost 14.30 dollars express $C$ as a function of $n$.

Problem 113

Coffee locations Suppose that in 2008 there were approximately $6,400$ of your coffee company locations. Suppose that in 2012 this number had grown to approximately $13,168$. Write a linear function that represents the number of coffee locations $n$ as a function of time $t .$ Let $t=0$ represent 2008.

Problem 114

Cliff divers The cliff divers of Acapulco amaze tourists with their diving skills. The velocity $v$ of a diver is a function of the time $t$ the diver has fallen. If the initial velocity of the diver is 2 feet per second and $v=-66$ feet per second when $t=2$ seconds, express $v$ as a function $t$

Problem 115

Exchange rates If fifty U.S. dollars can be exchanged for 69.5550 Euros and 125 U.S. dollars can be exchanged for 173.8875 Euros, write a linear function that represents the number of Euros $E$ in terms of U.S. dollars $D$

Problem 116

Exchange rates If fifty U.S. dollars can be exchanged for 600.1100 Mexican pesos and 125 U.S. dollars can be exchanged for 1500.275 Mexican pesos, write a linear function that represents the number of Mexican pesos $P$ in terms of U.S. dollars $D$

Problem 117

Find all values of $x$ that will make $f(x)=0$. $$f(x)=3 x+2$$

Problem 118

Find all values of $x$ that will make $f(x)=0$. $$f(x)=-2 x-5$$

Problem 119

Find all values of $x$ that will make $f(x)=0$. Write a paragraph explaining how to find the domain of a function.

Problem 120

Find all values of $x$ that will make $f(x)=0$. Write a paragraph explaining how to find the range of a function.

Problem 121

Find all values of $x$ that will make $f(x)=0$. Explain why all functions are relations, but not all relations are functions.

Problem 122

Find all values of $x$ that will make $f(x)=0$. Use a graphing calculator to graph the function $f(x)=\sqrt{x},$ and use $[TRACE]$ and $[ZOOM]$ to find $\sqrt{5}$ to three decimal places.

Problem 123

Consider this set: $\left\{-3,-1,0,0.5, \frac{3}{4}, 1, \pi, 7,8\right\}$. Which numbers are natural numbers?

Problem 124

Consider this set: $\left\{-3,-1,0,0.5, \frac{3}{4}, 1, \pi, 7,8\right\}$. Which numbers are rational numbers?

Problem 125

Consider this set: $\left\{-3,-1,0,0.5, \frac{3}{4}, 1, \pi, 7,8\right\}$. Which numbers are prime numbers?

Problem 126

Consider this set: $\left\{-3,-1,0,0.5, \frac{3}{4}, 1, \pi, 7,8\right\}$. Which numbers are even numbers?

Problem 127

Write each in interval notation. (FIGURE CANNOT COPY)

Problem 128

Problem 129.

Graph each union of two intervals. $$(-3,5) \cup[6, \infty)$$

Problem 130

Graph each union of two intervals. $$(-\infty, 0) \cup(0, \infty)$$