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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