algebra 2 1.1 homework relations and functions

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Unit 2: complex numbers, unit 3: polynomial factorization, unit 4: polynomial division, unit 5: polynomial graphs, unit 6: rational exponents and radicals, unit 7: exponential models, unit 8: logarithms, unit 9: transformations of functions, unit 10: equations, unit 11: trigonometry, unit 12: modeling.

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1.1.1: Relations and Functions

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Finding the Domain and Range of Functions

You just got a new part-time job at the mall that pays a base rate of $150/week plus $5/sale. Your boss encourages you to make as many sales as possible but she will cap your weekly earnings at $250. What are the domain and range of the function represented by this situation?

Domain and Range of a Function

The input and output of a function is also called the domain and range. The domain of a function is the set of all input values. The range of a function is the set of all output values. Sometimes, a function is a set of points. In this case, the domain is all the x− values and the range is all the y− values. Functions can also be linear and polynomial equations. In these instances, you need to graph the function to see where it is defined. You may notice that some functions are defined for “all real numbers.” The symbol, ℝ, is used to denote the set of all real numbers.

Let's determine if {(9, 2), (7, -3), (4, -6), (-10, 4), (-2, -7)} is a function. If so, we will find the domain and range.

First, this is a function because the x− values do not repeat. To find the domain, we need to list all the x− values. The range is all the y− values. Typically, you would list the values in the order they appear. Notice the notation.

x∈{9,7,4,−10,−2}

y∈{2,−3,−6,4,−7}

The ∈ symbol means “an element of/in.” The braces, { }, around the x and y−values, indicate that each is a set. In words, you would say, “x is an element in the set 9, 7, 4, -10, and 2.” Depending on the text, you may see “:” (colon) interchanged with the “∈” symbol and sets without { } around them.

Let's find the domain and range for the following problems.

• y=x−3

Because this is a linear equation we also know that it is a linear function. All lines continue forever in both directions, as indicated by the arrows.

Notice the line is solid, there are no dashes or breaks. This means that it is continuous. A continuous function has a value for every x, or the domain is all real numbers. Can you plug in ANY value for x and get a y−value? Yes. There are a few ways to write this.

Domain : x∈ℝ, x∈(−∞,∞), x is all reals

In words, x is an element in the set of real numbers.

The second option, (−∞,∞), is an interval, not a point. The parenthesis indicate that infinity, ∞, and negative infinity, −∞, are not included in the interval, but every number between them is. To include an endpoint in the interval, use [ or ] brackets. This is called interval notation.

The range of this function is also continuous. Therefore, the range is also the set of all real numbers. We can write the range in the same ways we wrote the domain, but with y instead of x.

Range : y∈ℝ or y∈(−∞,∞)

This is a function, even though it might not look like it. This type of function is called a piecewise function because it pieces together two or more parts of other functions.

To find the domain, look at the possible x−values. Notice that when x is between -2 and -1 it is not defined, or there are no x−values.

Mathematically, this would be written: x∈(−∞,−2]∪(−1,∞). The ∪ symbol means “ union .” In words, the domain is “all real numbers except those between -2 and -1.” Notice that -2 is included in the domain because the dot at -2 is closed. To find the range, we need to look at the possible y−values. Changing our viewpoint to look at the y−axis, at first glance, it looks like the function is not defined from 1 to -3.

However, upon further investigation, the branch on the left does pass through the yellow region, where we though the function was not defined. This means that the function is defined between 1 and -3 and thus for all real numbers. However, below -3, there are no y−values. The range is y∈[−3,∞).

Earlier, you were asked to find the domain and range of the function of your sales, where you make a base rate of $150/week plus $5/sale (your weekly earnings are capped at $250/week).

The function represented by this situation can be written as y=150+5x, where x is the number of sales you make. You can't make a negative number of sales, so the least amount of sales you can make is zero. To find the maximum number of sales before you reach the cap, we must plug in $250 for y .

Therefore, the domain of the function is 0≤x≤20.

To find the range, plug the two extremes of the domain into the equation. When x equals 0, y equals 150, and when x equals 20, y equals 250.

Therefore the range of the function is 150≤y≤250.

Find the domain and range of the following function: {(8, 3), (-4, 2), (-6, 1), (5, 7)}.

Domain: x∈{8,−4,−6,5} Range: y∈{3,2,1,7}

Find the domain and range of the following function: y=−\(\ 1\over 2\)x+4.

Domain: x∈ℝ Range: y∈ℝ

Find the domain and range of the following function.

This is a piecewise function. The x−values are not defined from -2 to 1. The range looks like it is not defined from 1 to 7, but the lines continue on, filling in that space as x gets larger, both negatively and positively.

Domain: x∈(−∞,−2)∪(1,∞) Range: y∈ℝ

This is a parabola, the graph of a quadratic equation. Even though it might not look like it, the ends of the graph continue up, infinitely, and x keeps growing. In other words, x is not limited to be between -9 and 5. It is all real numbers. The range, however, seems to start at -6 and is all real numbers above that value.

Domain: x∈ℝ Range: y∈[−6,∞)

Determine if the following sets of points are functions. If so, state the domain and range.

• {(5, 6), (-1, 5), (7, -3), (0, 9)}
• {(9, 8), (-7, 8), (-7, 9), (8, 8)}
• {(6, 2), (-5, 6), (-5, 2)}
• {(-1, 2), (-6, 3), (10, 7), (8, 11)}
• {(5, 7), (3, 7), (5, 8), (8, 1)}
• {(-3, -4), (-5, -6), (1, 2), (2, -6)}

Find the domain and range of the following functions.

• y=3x−7
• 6x−2y=10

14. Challenge

15. Writing Make a general statement about the domain and range of all linear functions. Use the proper notation.

Image Attributions

[Figure 1] Credit: Laura Guerin;CK-12 Foundation Source: CK-12 Foundation

[Figure 2] Credit: Laura Guerin;CK-12 Foundation;CK-12 Foundation, Larame Spence Source: CK-12 Foundation;Desmos Graphing Calculator

[Figure 3] Credit: Laura Guerin;CK-12 Foundation Source: CK-12 Foundation

[Figure 4] Credit: Laura Guerin;CK-12 Foundation Source: CK-12 Foundation

[Figure 5] Credit: Laura Guerin;CK-12 Foundation Source: CK-12 Foundation

[Figure 6] Credit: Laura Guerin;CK-12 Foundation Source: CK-12 Foundation

[Figure 7] Credit: Laura Guerin;CK-12 Foundation Source: CK-12 Foundation

[Figure 8] Credit: Laura Guerin;CK-12 Foundation Source: CK-12 Foundation

[Figure 9] Credit: Laura Guerin;CK-12 Foundation Source: CK-12 Foundation

[Figure 10] Credit: CK-12 Foundation, Larame Spence Source: Desmos Graphing Calculator

MATH 1111 - College Algebra: 2.2 Intro to Functions

• 1.1 Sets and Set Operations
• 1.2 Linear Equations and Inequalities
• 1.3 Systems of Linear Equations
• 1.4 Polynomials; Operations with Polynomials
• 1.5 Factoring Polynomials
• 1.6 Quadratic Equations
• 1.7 Rational Expressions and Equations
• 1.8 Complex Numbers
• 2.1 Cartesian Coordinates/Relations

2.2 Intro to Functions

• 2.3 Operations with Functions
• 2.4 Graph of Functions
• 3.1 Linear Functions
• 3.2 Quadratic Functions and Quadratic Inequalities
• 4.1 Finding Zeros of Polynomial Functions
• 4.2 Graphing Polynomial Functions
• 4.3 Rational Functions
• 4.4 Rational Inequalities
• 5.1 Composition of Functions
• 5.2 Inverse Functions
• 5.3 Introduction to Exponential and Logarithmic Functions

At the end of this section students will be able to:

• Determine whether a relation is a function
• Find the domain and range of a function
• Evaluate functions

Required Reading

1.3 Introduction to Functions

Stitz-Zeager College Algebra  - pages  43-47

1.4 Function Notation

Stitz-Zeager College Algebra  - pages  55-59

Practice Exercises

Introduction to Functions

Stitz-Zeager College Algebra  - pages 49-54

Answers to practice exercises can be found on pages 53-54.

Function Notation

Stitz-Zeager College Algebra  - pages 63-65

Answers to practice exercises can be found on pages 69-74.

Supplemental Resources

Introduction to Functions  (tutorial):  West Texas A&M University Virtual Math Lab (College Algebra Tutorial 30)

Finding the Domain of a Function:

Evaluating Functions:

• << Previous: 2.1 Cartesian Coordinates/Relations
• Next: 2.3 Operations with Functions >>
• Last Updated: Apr 2, 2024 2:52 PM
• URL: https://libguides.gcsu.edu/math1111

Algebra Function Worksheets with Answer Keys

Feel free to download and enjoy these free worksheets on functions and relations .Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key.

• Domain and Range (Algebra 1)
• Functions vs Relations (distinguish function from relation, state domain etc..) (Algebra 2)
• Evaluating Functions (Algebra 2)
• 1 to 1 Functions (Algebra 2)
• Composition of Functions (Algebra 2)
• Inverse Functions Worksheet (Algebra 2)
• Operations With Functions (Algebra 2)
• Functions Review Worksheet (Algebra 2)

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2.1E: Exercises - Relations and Functions

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Practice Makes Perfect

Find the Domain and Range of a Relation

In the following exercises, for each relation a. find the domain of the relation b. find the range of the relation.

1. \({\{(1,4),(2,8),(3,12),(4,16),(5,20)}\}\)

a. \({\{1, 2, 3, 4, 5}\}\) b. \({\{4, 8, 12, 16, 20}\}\)

2. \({\{(1,−2),(2,−4),(3,−6),(4,−8),(5,−10)}\}\)

3. \({\{(1,7),(5,3),(7,9),(−2,−3),(−2,8)}\}\)

a. \({\{1, 5, 7, −2}\}\) b. \({\{7, 3, 9, −3, 8}\}\)

4. \({\{(11,3),(−2,−7),(4,−8),(4,17),(−6,9)}\}\)

In the following exercises, use the mapping of the relation to a. list the ordered pairs of the relation, b. find the domain of the relation, and c. find the range of the relation.

a. (Rebecca, January 18), (Jennifer, April 1), (John, January 18), (Hector, June 23), (Luis, February 15), (Ebony, April 7), (Raphael, November 6), (Meredith, August 19), (Karen, August 19), (Joseph, July 30) b. {Rebecca, Jennifer, John, Hector, Luis, Ebony, Raphael, Meredith, Karen, Joseph} c. {January 18, April 1, June 23, February 15, April 7, November 6, August 19, July 30}

7. For a woman of height \(5'4''\) the mapping below shows the corresponding Body Mass Index (BMI). The body mass index is a measurement of body fat based on height and weight. A BMI of \(18.5–24.9\) is considered healthy.

a. \((+100, 17. 2), (110, 18.9), (120, 20.6), (130, 22.3), (140, 24.0), (150, 25.7), (160, 27.5)\) b. \({\{+100, 110, 120, 130, 140, 150, 160,}\}\) c. \({\{17.2, 18.9, 20.6, 22.3, 24.0, 25.7, 27.5}\}\)

8. For a man of height \(5'11''\) the mapping below shows the corresponding Body Mass Index (BMI). The body mass index is a measurement of body fat based on height and weight. A BMI of \(18.5–24.9\) is considered healthy.

In the following exercises, use the graph of the relation to a. list the ordered pairs of the relation b. find the domain of the relation c. find the range of the relation.

a. \((2, 3), (4, −3), (−2, −1), (−3, 4), (4, −1), (0, −3)\) b. \({\{−3, −2, 0, 2, 4}\}\) c. \({\{−3, −1, 3, 4}\}\)

a. \((1, 4), (1, −4), (−1, 4), (−1, −4), (0, 3), (0, −3)\) b. \({\{−1, 0, 1}\}\) c. \({\{−4, −3, 3,4}\}\)

Determine if a Relation is a Function

In the following exercises, use the set of ordered pairs to a. determine whether the relation is a function, b. find the domain of the relation, and c. find the range of the relation.

13. \( {\{(−3,9),(−2,4),(−1,1), (0,0),(1,1),(2,4),(3,9)}\}\)

a. yes b. \({\{−3, −2, −1, 0, 1, 2, 3}\}\) c. \({\{9, 4, 1, 0}\}\)

14. \({\{(9,−3),(4,−2),(1,−1),(0,0),(1,1),(4,2),(9,3)}\}\)

15. \({\{(−3,27),(−2,8),(−1,1), (0,0),(1,1),(2,8),(3,27)}\}\)

a. yes b. \({\{−3, −2, −1, 0, 1, 2, 3}\}\) c. \({\{0, 1, 8, 27}\}\)

16. \({\{(−3,−27),(−2,−8),(−1,−1), (0,0),(1,1),(2,8),(3,27)}\}\)

In the following exercises, use the mapping to a. determine whether the relation is a function, b. find the domain of the function, and c. find the range of the function.

a. yes b. \({\{−3, −2, −1, 0, 1, 2, 3}\}\) c. \({\{0, 1, 2, 3}\}\)

a. no b. {Jenny, R and y, Dennis, Emily, Raul} c. {RHern and [email protected] , [email protected] , [email protected] , [email protected] , [email protected] , [email protected] , R and [email protected] }

In the following exercises, determine whether each equation is a function.

21. a. \(2x+y=−3\) b. \(y=x^2\) c. \(x+y^2=−5\)

a. yes b. yes c. no

22. a. \(y=3x−5\) b. \(y=x^3\) c. \(2x+y^2=4\)

23. a. \(y−3x^3=2\) b. \(x+y^2=3\) c. \(3x−2y=6\)

a. yes b. no c. yes

24. a. \(2x−4y=8\) b. \(−4=x^2−y\) c. \(y^2=−x+5\)

Find the Value of a Function

In the following exercises, evaluate the function: a. \(f(2)\) b. \(f(−1)\) c. \(f(a)\).

25. \(f(x)=5x−3\)

a. \(f(2)=7\) b. \(f(−1)=−8\) c. \(f(a)=5a−3\)

26. \(f(x)=3x+4\)

27. \(f(x)=−4x+2\)

a. \(f(2)=−6\) b. \(f(−1)=6\) c. \(f(a)=−4a+2\)

28. \(f(x)=−6x−3\)

29. \(f(x)=x^2−x+3\)

a. \(f(2)=5\) b. \(f(−1)=5\) c. \(f(a)=a^2−a+3\)

30. \(f(x)=x^2+x−2\)

31. \(f(x)=2x^2−x+3\)

a. \(f(2)=9\) b. \(f(−1)=6\) c. \(f(a)=2a^2−a+3\)

32. \(f(x)=3x^2+x−2\)

In the following exercises, evaluate the function: a. \(g(h^2)\) b. \(g(x+2)\) c. \(g(x)+g(2)\).

33. \(g(x)=2x+1\)

a. \(g(h^2)=2h^2+1\) b. \(g(x+2)=4x+5\) c. \(g(x)+g(2)=2x+6\)

34. \(g(x)=5x−8\)

35. \(g(x)=−3x−2\)

a. \(g(h^2)=−3h^2−2\) b. \(g(x+2)=−3x−8\) c. \(g(x)+g(2)=−3x−10\)

36. \(g(x)=−8x+2\)

37. \(g(x)=3−x\)

a. \(g(h^2)=3−h^2\) b. \(g(x+2)=1−x\) c. \(g(x)+g(2)=4−x\)

38. \(g(x)=7−5x\)

In the following exercises, evaluate the function.

39. \(f(x)=3x^2−5x\); \(f(2)\)

40. \(g(x)=4x^2−3x\); \(g(3)\)

41. \(F(x)=2x^2−3x+1\); \(F(−1)\)

42. \(G(x)=3x^2−5x+2\); \(G(−2)\)

43. \(h(t)=2|t−5|+4\); \(f(−4)\)

44. \(h(y)=3|y−1|−3\); \(h(−4)\)

45. \(f(x)=x+2x−1\); \(f(2)\)

46. \(g(x)=x−2x+2\); \(g(4)\)

In the following exercises, solve.

47. The number of unwatched shows in Sylvia’s DVR is 85. This number grows by 20 unwatched shows per week. The function \(N(t)=85+20t\) represents the relation between the number of unwatched shows, N , and the time, t , measured in weeks.

a. Determine the independent and dependent variable.

b. Find \(N(4)\). Explain what this result means

a. t IND; N DEP b. \(N(4)=165\) the number of unwatched shows in Sylvia’s DVR at the fourth week.

48. Every day a new puzzle is downloaded into Ken’s account. Right now he has 43 puzzles in his account. The function \(N(t)=43+t\) represents the relation between the number of puzzles, N , and the time, t , measured in days.

b. Find \(N(30)\). Explain what this result means.

49. The daily cost to the printing company to print a book is modeled by the function \(C(x)=3.25x+1500\) where C is the total daily cost and x is the number of books printed.

b. Find \(N(0)\). Explain what this result means.

c. Find \(N(1000)\). Explain what this result means.

a. x IND; C DEP b. \(N(0)=1500\) the daily cost if no books are printed c. \(N(1000)=4750\) the daily cost of printing 1000 books

50. The daily cost to the manufacturing company is modeled by the function \(C(x)=7.25x+2500\) where \(C(x)\) is the total daily cost and x is the number of items manufactured.

b. Find \(C(0)\). Explain what this result means.

c. Find \(C(1000)\). Explain what this result means.

Writing Exercises

51. In your own words, explain the difference between a relation and a function.

52. In your own words, explain what is meant by domain and range.

53. Is every relation a function? Is every function a relation?

54. How do you find the value of a function?

a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

b. After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?