Download Chapter 2 Functions - Missouri Western State University

MAT116 Final Review
Session Chapter 2:
Functions and Graphs
Note: Always give exact answers and always put your
in interval notation when applicable.
answers
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?
3.
4.
Fido
Bossy
Silver
Frisky
Polly
Civil War
450
550
2
40
8
3
1963
WWI
1950
WWII
1939
Korean
1917
Vietnam
1861
5.
x
-1
1
5
7
10
15
y
4
-3
9
4
-3
9
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 − 3𝑥 2 = 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 − 𝑥 2
13. 𝑦 = 16 + 𝑥 2
14. 𝑦 2 = 16 − 𝑥 2
Circles
• A graph of any equation of the form (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 =
𝑟 2 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 Transformations
There 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 +
20.
𝑛 𝑥 = (𝑥 + 2)2
5
2
5
|
2
21.
𝑞 𝑥 = |𝑥 −
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.
Examples: Graph the following
on your calculator.
23.
𝑦 = 𝑥2
1 2
𝑥
3
24.
𝑦=
25.
𝑦 = 5𝑥 2
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 = 2𝑥 2 − 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 = 𝑥 2 − 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 11.
• 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)
2)
3)
4)
5)
Replace f(x) with y
Interchange x and y
Solve the equation for y.
Replace y with f -1(x).
Verify that Df = Rf-1 and 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 = 𝑥 2 + 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.
37.
x
y
2
0
3
1
6
2
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
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1) A function
2) Not a function
3) Not a function
4) A function
5) A function
6) Domain: (-inf, inf), Range: [-1, inf)
7) Domain: (-inf, inf), Range: (-inf, inf)
8) Domain: (-inf, inf), Range: [-3, inf)
9) Domain: (-inf, -3], Range: [0, inf)
10) Domain: (-inf, inf), Range: (-inf, 1]
11) Domain: [0, inf), Range: [-3, inf)
12) Yes
13) Yes
14) Not a function
15) Center = (2, -4) Radius = 4
16) Increasing on [-3, -1]U[0,2]
Decreasing on [2,3]
Constant [-1,0]
• 17) 2 units right
• 18) 2 units down
19) 5/2 units up
20) 2 units left
21) 5/2 units right
22) 7 units right, 3 units up
23 – 25) Graph
26) D: (-inf, inf) R: (-inf, 1]
27) Magnified by 2, moved right 2 units, up 4 units
28) Flipped over x-axis, moved 3 units right, move
down ½ units
29) 5
30) -3
31) 348
32) x ≠ 2
33) 9𝑥 2 − 24𝑥 − 17
34) 3𝑥 2 − 7
𝑥+3
35) y =
2
36) Graph