MAT116 Final Review Session Chapter 2:

Functions and Graphs

Note: Always give exact answers and always put your answers

in interval notation when applicable.

Section 1

If the value of x determines the value of y, we say that “y is a function of x.”

If there is more than one value of y corresponding to a particular x-value then y is not determined by x.

• (i.e., y is NOT a function of x)

Vertical Line Test

A graph is a graph of a function if and only if there is no vertical line that passes through the graph more than once.

Examples: Do these represent a function?1. 2.

Examples: Do these relations represent a function?

x -1 1 5 7 10 15

y 4 -3 9 4 -3 9

Fido

BossySilverFriskyPolly

45055024083

Civil War

WWI

WWII

Korean

Vietnam

1963

1950

1939

1917

1861

3. 4.

5.

Domain and Range

• The set of all possible x-values is defined as the domain.

• The set of all resulting y-values is defined as the range.

Examples: Determine if the following are functions, state their domain and range.

6. 3𝑦 − 3𝑥2 = 12𝑥 + 9

7. 𝑦 = 3𝑥 + 9

8. 𝑦 = 𝑥 − 3

9. 𝑦 = −3 − 𝑥

1-1

• A function is 1-1 if and only if it’s graph passes the vertical line test AND the horizontal line test.

Examples: Graph the function, state if it is 1-1, the domain and range

10. 𝑦 = 1 − 𝑥2

11. 𝑦 = 𝑥 − 3

Examples: Determine if the following equations are functions.

12. 𝑦 = 16 − 𝑥2

13. 𝑦 = 16 + 𝑥2

14. 𝑦2 = 16 − 𝑥2

Circles

• A graph of any equation of the form (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 =𝑟2 is a circle with center (ℎ, 𝑘) and radius 𝑟.

• Circles do not represent a function.

Examples: What is the center and radius of the following circle?

15. (𝑦 − 2)2+(𝑥 + 4)2= 16

Examples: Determine where the graph is increasing, decreasing or constant.

16.

Transformations

• There are two categories of transformations:

• Rigid Transformations

• Nonrigid Transformations

Rigid TransformationsThere are 3 different rigid transformations:

1. Vertical – Shifts up and down• f(x) + a is f(x) shifted upward a units

• f(x) – a is f(x) shifted downward a units

2. Horizontal – Shifts left and right• f(x + a) is f(x) shifted left a units

• f(x – a) is f(x) shifted right a units

3. Reflection – reflects over and axis

• –f(x) is f(x) flipped upside down (reflected over x-axis)

Examples: How many units is each function shifted? In which direction?

17. ℎ 𝑥 = 𝑥 − 2

18. 𝑓 𝑥 = 𝑥2 − 2

19. 𝑔 𝑥 = 𝑥 +5

2

20. 𝑛 𝑥 = (𝑥 + 2)2

21. 𝑞 𝑥 = |𝑥 −5

2|

22. 𝑝 𝑥 = (𝑥 − 7)5+3

Nonrigid Transformations

There are 2 types of nonrigid transformations.

1. Stretching

• Let 𝑎 > 1. Then 𝑦 = 𝑎 ∙ 𝑓(𝑥) stretches the graph by a factor of a.

2. Shrinking

• Let 0 < 𝑎 < 1. Then 𝑦 = 𝑎 ∙ 𝑓(𝑥) shrinks the graph by a factor of a.

* All the y-coordinates on f(x) are multiplied by a, so the graph stretches or shrinks in the y direction.

Examples: Graph the following on your calculator.

23. 𝑦 = 𝑥2

24. 𝑦 =1

3𝑥2

25. 𝑦 = 5𝑥2

Examples: Use transformations to graph the following function. State the domain and range.

26. 𝑦 = − 𝑥 − 2 2 + 1

Note: Be sure to follow the order of operations while translating the function. “Please Excuse My Dear Aunt Sally.”

(Parentheses, exponents, multiplication/division, addition/subtraction).

Examples: Describe the transformation in words.

27. 𝑡 𝑥 = 2 𝑥 − 2 + 4

28. 𝑓 𝑥 = − 𝑥 − 3 −1

2

Operations with Functions

• 𝑓 + 𝑔 𝑥 = 𝑓 𝑥 + 𝑔 𝑥

• 𝑓 − 𝑔 𝑥 = 𝑓 𝑥 − 𝑔 𝑥

• 𝑓 ∙ 𝑔 𝑥 = 𝑓 𝑥 ∙ 𝑔 𝑥

• 𝑓/𝑔 𝑥 = 𝑓(𝑥)/𝑔(𝑥) where 𝑔(𝑥) ≠ 0

Examples: Evaluate the following.

Let 𝑦 𝑥 = 2𝑥2 − 3 and 𝑤 𝑥 = 2𝑥 + 4.

29. (𝑦 + 𝑤)(1)

30. (𝑤 − 𝑦)(2)

31. (𝑦 ∙ 𝑤)(4)

32. 𝑦/𝑤(𝑥)

Composition

If f and g are two functions, the composition of f and g, written f ∘ g, is defined as follows:

Examples: Evaluate the following.

Let 𝑓 𝑥 = 𝑥2 − 1 and 𝑔 𝑥 = 3𝑥 − 4.

33. (𝑓 ∘ 𝑔)(𝑥)

34. (𝑔 ∘ 𝑓)(𝑥)

Inverse Functions• A function has an inverse if and only if the function is 1-1.

• The inverse of a one-to-one function f(x) is the function 𝑓−1 such that:

• Note: The domain of 𝑓(𝑥) is the range of 𝑓−1(𝑥)

The range of 𝑓(𝑥) is the domain of 𝑓−1(𝑥)

To find the inverse of a function f(x):

1) Replace 𝑓(𝑥) with 𝑦

2) Interchange 𝑥 and 𝑦

3) Solve the equation for 𝑦.

4) Replace y with 𝑓−1(𝑥).

5) Verify that 𝐷𝑓 = 𝑅𝑓−1and vice versa.

Examples: Find the equation of the inverse.

35. 𝑓 𝑥 = 2𝑥 − 3

Examples: Graph the inverse of the following function:

36. 𝑓 𝑥 = 𝑥2 + 6𝑥 + 9; 𝑥 ≥ −3

Remember: reflect the graph of f(x) over the line 𝑦 = 𝑥 to get the graph of the inverse.

Examples: Find the inverse.

x y

2 0

3 1

6 2

37.

Chapter 2 Review

• Determine if it’s a function

• Graphs of functions

• Finding Domain and Range

• Operations of Functions

• Transformations

• Functions and their Inverses

Example Answers• 1) A function• 2) Not a function• 3) Not a function• 4) A function• 5) A function• 6) Domain: (−∞,∞), Range: [−1,∞)• 7) Domain: (−∞,∞), Range: (−∞,∞)• 8) Domain: (−∞,∞), Range: [−3,∞)• 9) Domain: (−∞,−3], Range: [0,∞)• 10) Domain: (−∞,∞), Range: (−∞, 1]• 11) Domain: [0,∞), Range: [−3,∞)• 12) Yes• 13) Yes• 14) Not a function• 15) Center = (2, -4) Radius = 4• 16) Increasing on [−3,−1] ∪ [0,2]

Decreasing on [2,3]Constant [−1,0]

• 17) 2 units right• 18) 2 units down

19) 5

2units up

20) 2 units left

21) 5

2units right

22) 7 units right, 3 units up23 – 25) Graph 26) D: (−∞,∞) R: (−∞, 1]27) Magnified by 2, moved right 2 units, up 4 units28) Flipped over x-axis, moved 3 units right, move

down ½ units29) 530) 331) 46832) x ≠ −233) 9𝑥2 − 24𝑥 − 1734) 3𝑥2 − 7

35) y =𝑥+3

2

36) Graph