Inverse Functions

Show your work for each problem.

1.
If , what is ?
1. a)
2. b)
3. c)
4. d)
Answer:
If , then because the inverse reverses the function. What goes in comes out, and what comes out goes in.
2.
Which test determines if a function has an inverse?
1. a)Origin test
2. b)Vertical line test
3. c)Symmetry test
4. d)Horizontal line test
Answer: Horizontal line test
The horizontal line test determines if a function is one-to-one. If any horizontal line crosses the graph more than once, the function fails the test and has no inverse.
3.
The graph of is the reflection of over which line?
1. a)
2. b)
3. c)
4. d)
Answer:
Since the inverse swaps x and y values, the point on becomes on . This is exactly what reflection over the line does.
4.
Which function does NOT have an inverse?
1. a)
2. b)
3. c)
4. d)
Answer:
does not have an inverse because . Two different inputs give the same output, so it fails the horizontal line test. The other functions are all one-to-one.
5.
If , what is ?
1. a)
2. b)
3. c)
4. d)
Answer:
Start with . Swap: . Add 8: . Divide by 4: . So .
6.
If , what is ?
Answer: 5
Since , the inverse function takes 12 back to 5. Therefore, .
7.
Find if . Write your answer in the form
Answer: 7
If adds 7, then subtracts 7 to undo it. The answer is 7.
8.
If and , calculate .
Answer: 9
First, . Then . Since and are inverses, composing them returns the original input.
9.
Find if .
Answer: 3
The inverse is . So .
10.
If , find .
Answer: 10
Start with . Swap: . Multiply by 3: . Subtract 2: . So and .
11.
The domain of equals the range of . If with domain , what is the domain of ? (Answer: ?)
Answer: 0
When , produces outputs (the range). Since the domain of equals the range of , the domain of is .
12.
Find the inverse of
Answer: x + 4
1. Write . Now swap x and y. What equation do you get? x = y - 4
2. Solve for y by adding 4 to both sides: y = x + 4
3. Write the inverse function: x + 4
13.
Find the inverse of
Answer: (x - 10) / 5
1. Write . After swapping x and y, what equation do you have? x = 5y + 10
2. Subtract 10 from both sides: x - 10 = 5y
3. Divide both sides by 5 to solve for y: (x - 10) / 5
4. Write = (x - 10) / 5
14.
Verify that and are inverses by showing
Answer: x
1. Write : (x + 4) / 2
2. Substitute into : . Simplify : x + 4
3. Now subtract 4: x
4. Therefore, x
15.
Find the inverse of
Answer: 3(x - 2)
1. Write . After swapping, the equation is: x = y/3 + 2
2. Subtract 2 from both sides: x - 2 = y/3
3. Multiply both sides by 3: 3(x - 2) = y
4. So 3(x - 2)
16.
Find the inverse of , where
Answer: (-2x - 1) / (x - 1)
1. Write . After swapping x and y: x = (y - 1)/(y + 2)
2. Multiply both sides by : x(y + 2) = y - 1
3. Expand the left side: xy + 2x = y - 1
4. Move all y terms to one side: -2x - 1
5. Factor out y: . Solve for y: (-2x - 1)/(x - 1)
17.
If is on the graph of , find the corresponding point on the graph of
Answer: (9, 4)
1. If is on , what does this tell us about ? f(4) = 9
2. Since , what is ? 4
3. So the point on is: (9, 4)
18.
Find the inverse of where
Answer: x^2 - 4
1. Write . After swapping: x = sqrt(y + 4)
2. Square both sides to eliminate the square root: x^2 = y + 4
3. Solve for y by subtracting 4: y = x^2 - 4
4. What is the domain restriction for ? 0