# Chapter 1: Expressions, Equations, and Functions

**Basic**Created by: CK-12

## Introduction

The study of expressions, equations, and functions is the basis of mathematics. Each mathematical subject requires knowledge of manipulating equations to solve for a variable. Careers such as automobile accident investigators, quality control engineers, and insurance originators use equations to determine the value of variables.

Functions are methods of explaining relationships and can be represented as a rule, a graph, a table, or in words. The amount of money in a savings account, how many miles run in a year, or the number of trout in a pond are all described using functions.

Throughout this chapter, you will learn how to choose the best variables to describe a situation, simplify an expression using the Order of Operations, describe functions in various ways, write equations, and solve problems using a systematic approach.

## Chapter Outline

- 1.1. Variable Expressions
- 1.2. Expressions with One or More Variables
- 1.3. PEMDAS
- 1.4. Algebra Expressions with Fraction Bars
- 1.5. Calculator Use with Algebra Expressions
- 1.6. Patterns and Expressions
- 1.7. Words that Describe Patterns
- 1.8. Equations that Describe Patterns
- 1.9. Inequalities that Describe Patterns
- 1.10. Function Notation
- 1.11. Domain and Range of a Function
- 1.12. Functions that Describe Situations
- 1.13. Functions on a Cartesian Plane
- 1.14. Vertical Line Test
- 1.15. Problem-Solving Models
- 1.16. Trends in Data

### Chapter Summary

## Summary

This chapter first deals with expressions and how to evaluate them by using the correct order of operations. Tips on using a calculator are also given. It then builds on this knowledge by moving on to talk about equations and inequalities and the methods used to solve them. Next, functions are discussed in detail, with instruction given on using the proper notation, determining a function's domain and range, and graphing a function. How to determine whether or not a relation is a function is also covered. Finally, the chapter concludes by highlighting some general problem-solving strategies.

### Expressions, Equations, and Functions Review

Define the following words:

- Domain
- Range
- Solution
- Evaluate
- Substitute
- Operation
- Variable
- Algebraic expression
- Equation
- Algebraic inequality
- Function
- Independent variable

Evaluate the following expressions.

- \begin{align*}3y (7 - (z - y))\end{align*}; use \begin{align*}y = -7\end{align*} and \begin{align*}z = 2\end{align*}
- \begin{align*}\frac{m + 3n - p}{4}\end{align*}; use \begin{align*}m = 9, \ n = 7,\end{align*} and \begin{align*}p = 2\end{align*}
- \begin{align*}|p| - \left (\frac{n}{2} \right )^3\end{align*}; use \begin{align*}n = 2\end{align*} and \begin{align*}p = 3\end{align*}
- \begin{align*}|v-21|\end{align*}; use \begin{align*}v=-70\end{align*}

Choose an appropriate variable to describe the situation.

- The number of candies you can eat in a day
- The number of tomatoes a plant can grow
- The number of cats at a humane society
- The amount of snow on the ground
- The number of water skiers on a lake
- The number of geese migrating south
- The number of people at a trade show

The surface area of a sphere is found by the formula \begin{align*}A = 4 \pi r^2\end{align*}. Determine the surface area for the following radii/diameters.

- \begin{align*}radius=10 \ inches\end{align*}
- \begin{align*}radius=2.4 \ cm\end{align*}
- \begin{align*}diameter=19 \ meters\end{align*}
- \begin{align*}radius=0.98 \ mm\end{align*}
- \begin{align*}diameter=5.5 \ inches\end{align*}

Insert parentheses to make a true equation.

- \begin{align*}1 + 2 \cdot 3 + 4 = 15\end{align*}
- \begin{align*}5 \cdot 3 - 2 + 6 = 35\end{align*}
- \begin{align*}3 + 1 \cdot 7 - 2^2 \cdot 9 - 7 = 24\end{align*}
- \begin{align*}4 + 6 \cdot 2 \cdot 5 - 3 = 40\end{align*}
- \begin{align*}3^2 + 2 \cdot 7 - 4 = 33\end{align*}

Translate the following into an algebraic expression, equation, or inequality.

- Thirty-seven more than a number is 612.
- The product of \begin{align*}u\end{align*} and –7 equals 343.
- The quotient of \begin{align*}k\end{align*} and 18
- Eleven less than a number is 43.
- A number divided by –9 is –78.
- The difference between 8 and \begin{align*}h\end{align*} is 25.
- The product of 8, –2, and \begin{align*}r\end{align*}
- Four plus \begin{align*}m\end{align*} is less than or equal to 19.
- Six is less than \begin{align*}c\end{align*}.
- Forty-two less than \begin{align*}y\end{align*} is greater than 57.

Write the pattern shown in the table with words and with an algebraic equation.

- \begin{align*}&\text{Movies watched} && 0 && 1 && 2 && 3 && 4 && 5\\ &\text{Total time} && 0 && 1.5 && 3 && 4.5 && 6 && 7.5\end{align*}
- A case of donuts is sold by the half-dozen. Suppose 168 people purchase cases of donuts. How many individual donuts have been sold?
- Write an inequality to represent the situation:
*Peter’s Lawn Mowing Service charges $10 per mowing job and $35 per landscaping job. Peter earns at least $8,600 each summer.*

Check that the given number is a solution to the given equation or inequality.

- \begin{align*}t=0.9, \ 54 \le 7(9t+5)\end{align*}
- \begin{align*}f=2; \ f+2+5f = 14\end{align*}
- \begin{align*}p=-6; \ 4p-5p \le 5\end{align*}
- Logan has a cell phone service that charges $18 dollars per month and $0.05 per text message. Represent Logan’s monthly cost as a function of the number of texts he sends per month.
- An online video club charges $14.99 per month. Represent the total cost of the video club as a function of the number of months that someone has been a member.
- What is the domain and range for the following graph?
- Henry invested $5,100 in a vending machine service. Each machine pays him $128. How many machines does Henry need to install to break even?
- Is the following relation a function?

Solve the following questions using the 4-step problem-solving plan.

- Together, the Raccoons and the Pelicans won 38 games. If the Raccoons won 13 games, how many games did the Pelicans win?
- Elmville has 250 fewer people than Maplewood. Elmville has 900 people. How many people live in Maplewood?
- The cell phone Bonus Plan gives you 4 times as many minutes as the Basic Plan. The Bonus Plan gives you a total of 1200 minutes. How many minutes does the Basic Plan give?
- Margarite exercised for 24 minutes each day for a week. How many total minutes did Margarite exercise?
- The downtown theater costs $1.50 less than the mall theater. Each ticket at the downtown theater costs $8. How much do tickets at the mall theater cost?
- Mega Tape has 75 more feet of tape than everyday tape. A roll of Mega Tape has 225 feet of tape. How many feet does everyday tape have?
- In bowling DeWayne got 3.5 times as many strikes as Junior. If DeWayne got 28 strikes, how many strikes did Junior get?

### Expressions, Equations, and Functions Test

- Write the following as an algebraic equation and determine its value.
*On the stock market, Global First hit a price of $255 on Wednesday. This was $59 greater than the price on Tuesday. What was the price on Tuesday?* - The oak tree is 40 feet taller than the maple. Write an expression that represents the height of the oak.
- Graph the following ordered pairs: (1, 2), (2, 3), (3, 4), (4, 5) (5, 6), (6, 7).
- Determine the domain and range of the following function:
- Is the following relation a function? Explain your answer. \begin{align*}\left \{(3, 2), (3, 4), (5, 6), (7, 8) \right \}\end{align*}
- Evaluate the expression \begin{align*}(5bc)-a\end{align*} if \begin{align*}a=2,\ b=3, \ \text{and } \ c=4\end{align*}.
- Simplify: \begin{align*}3[36 \div (3+6)]\end{align*}.
- Translate the following into an algebraic equation and find the value of the variable:
*One-eighth of a pizza costs $1.09. How much was the entire pizza?* - Use the 4-step problem-solving method to determine the solution to the following:
*The freshman class has 17 more girls than boys. There are 561 freshmen. How many are girls?* - Underline the math verb in this sentence: 8 divided by \begin{align*}y\end{align*} is 48.
- Jesse packs 16 boxes per hour. Complete the table to represent this situation.

\begin{align*}&\text{Hours} && 0 && 2 && 4 && 5 && 8 && 10 && 12 && 14\\ &\text{Boxes}\end{align*}

- A group of students are in a room. After 18 leave, it is found that \begin{align*}\frac{7}{8}\end{align*} of the original number of students remain. How many students were in the room in the beginning?
- What are the domain and range of the following relation: \begin{align*}\left \{(2, 3), (4, 5), (6, 7), (-2, -3), (-3, -4) \right \}?\end{align*}
- Write a function rule for the following table:

\begin{align*}& \text{Time in hours,}\ x && 0 && 1 && 2 && 3 && 4\\ & \text{Distance in miles,}\ y && 0 && 60 && 120 && 180 && 240\end{align*}

- Determine if the given number is a solution to the following inequality: \begin{align*}\frac{6 - y}{y} > -8; \ y = 6\end{align*}

#### Texas Instruments Resources

*In the CK-12 Texas Instruments Algebra I FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9611.*