HW 1.2.6-7: Inverse of a Function
1. Using the table below
t (minutes)
f(t) (miles)
30
20
50
40
60
50
70
60
90
70
Find and interpret the following
a. f (60)
b. f
1
(60)
2. A function g(x) is given as a graph below. Find g (3) and g 1 (3)
3. Find a formula for the inverse function that gives Fahrenheit temperature given a Celsius
5
temperature. If C  h( F )  ( F  32) then, F = h -1 (C) =
9
4. Given the graph of f(x) shown, sketch a graph of f
1
( x) .
Assume that the function f is a one-to-one function.
5. If f (6)  7 , find f 1 (7)
6. If f (3)  2 , find f 1 (2)
7. If f 1  4   8 , find f (8)
8. If f 1  2  1 , find f ( 1)
David Lippman and Melonie Rasmussen © 2015. Precalculus: An Investigation of Functions Ed. 1.5.
Retrieved from: http://www.opentextbookstore.com/precalc/
9. If f  5  2 , find  f  5  
10. If f 1  4 , find  f 1 
1
1
11. Using the graph of f ( x) shown
a. Find f  0 
b. Solve f ( x)  0
c. Find f 1  0
d. Solve f 1  x   0
e. Sketch f -1 ( x )
12. Using the graph shown
a. Find g (1)
b. Solve g ( x)  1
c. Find g 1 (1)
d. Solve g 1  x   1
e. Sketch f -1 (x)
13. Use the table below to find the indicated quantities.
x
f(x)
0
8
1
0
2
7
3
4
4
2
5
6
6
5
7
3
8
9
9
1
a. Find f 1
b. Solve f ( x)  3
c. Find f 1  0
d. Solve f 1  x   7
14. Use the table below to fill in the missing values.
t
h(t)
0
6
1
0
2
1
3
7
4
2
5
3
6
5
7
4
8
9
a. Find h  6 
b. Solve h(t )  0
c. Find h1  5
d. Solve h1  t   1
David Lippman and Melonie Rasmussen © 2015. Precalculus: An Investigation of Functions Ed. 1.5.
Retrieved from: http://www.opentextbookstore.com/precalc/
For each table below, create a table for f 1  x  .
15. x
16.
3 6 9 13 14
f(x) 1 4 7 12 16
3 5 7 13 15
x
f(x) 2 6 9 11 16
For each function below, find f 1 ( x)
17. f  x   x  3
18. f  x   x  5
19. f  x   2 – x
20. f  x   3  x
21. f  x   11x  7
22 f  x   9  10 x
For each function, find a domain on which f is one-to-one and non-decreasing, then find the inverse of
f restricted to that domain.
2
2
23. f  x    x 7 
24. f  x    x  6 
25. f  x   x 2  5
26. f  x   x2  1
27. If f  x   x3  5 and g ( x)  3 x  5 , find
a. f ( g ( x))
b.
g ( f ( x))
c. What does this tell us about the relationship between f ( x) and g ( x ) ?
a.
2x
x
and g ( x) 
, find
1 x
2 x
f ( g ( x))
b.
g ( f ( x))
28. If f ( x) 
c. What does this tell us about the relationship between f ( x) and g ( x ) ?
David Lippman and Melonie Rasmussen © 2015. Precalculus: An Investigation of Functions Ed. 1.5.
Retrieved from: http://www.opentextbookstore.com/precalc/
Selected Answers:
5. The definition of the inverse function is the function that reverses the input and output. So if the
output is 7 when the input is 6, the inverse function 𝑓 −1 (𝑥) gives an output of 6 when the input is 7.
So, 𝑓 −1 (7) = 6.
7. The definition of the inverse function is the function which reverse the input and output of the
original function. So if the inverse function 𝑓 −1 (𝑥) gives an output of −8 when the input is −4, the
original function will do the opposite, giving an output of −4 when the input is −8. So 𝑓(−8) = −4.
−1
1
1
9. 𝑓(5) = 2, so (𝑓(5)) = (2)−1 = 21 = 2.
11.
(a) 𝑓(0) = 3
(b) Solving 𝑓(𝑥) = 0 asks the question: for what input is the output 0? The answer is 𝑥 = 2. So,
𝑓(2) = 0.
(c) This asks the same question as in part (b). When is the output 0? The answer is 𝑓 −1 (0) = 2.
(d) The statement from part (c) 𝑓 −1 (0) = 2 can be interpreted as “in the original function 𝑓(𝑥),
when the input is 2, the output is 0” because the inverse function reverses the original function. So, the statement 𝑓 −1 (𝑥) = 0 can be interpreted as “in the original function 𝑓(𝑥), when the
input is 0, what is the output?” the answer is 3. So, 𝑓 −1 (3) = 0.
13.
(a) 𝑓(1) = 0
(b) 𝑓(7) = 3
(c) 𝑓 −1 (0) = 1
(d) 𝑓 −1 (3) = 7
15.
𝑥
−1
𝑓 (𝑥)
1
3
4
6
7
9
12
13
16
14
17. The inverse function takes the output from your original function and gives you back the input, or
undoes what the function did. So if 𝑓(𝑥) adds 3 to 𝑥, to undo that, you would subtract 3 from 𝑥. So,
𝑓 −1 (𝑥) = 𝑥 − 3.
19. In this case, the function is its own inverse, in other words, putting an output back into the function
gives back the original input. So, 𝑓 −1 (𝑥) = 2 − 𝑥.
21. The inverse function takes the output from your original function and gives you back the input, or
undoes what the function did. So if 𝑓(𝑥) multiplies 11 by 𝑥 and then adds 7, to undo that, you would
𝑥−7
subtract 7 from 𝑥, and then divide by 11. So, 𝑓 −1 (𝑥) = 11 .
23. This function is one-to-one and non-decreasing on the interval 𝑥 > −7. The inverse function,
restricted to that domain, is 𝑓 −1 (𝑥) = √𝑥 − 7.
25. This function is one-to-one and non-decreasing on the interval 𝑥 > 0. The inverse function,
restricted to that domain, is 𝑓 −1 (𝑥) = √𝑥 + 5.
David Lippman and Melonie Rasmussen © 2015. Precalculus: An Investigation of Functions Ed. 1.5.
Retrieved from: http://www.opentextbookstore.com/precalc/
3
3
27. (a) 𝑓(𝑔(𝑥)) = (( √𝑥 + 5)) − 5, which just simplifies to 𝑥.
3
(b) 𝑔(𝑓(𝑥)) = ( √(𝑥 3 − 5) + 5), which just simplifies to 𝑥.
(c) This tells us that 𝑓(𝑥) and 𝑔(𝑥) are inverses, or, they undo each other.
David Lippman and Melonie Rasmussen © 2015. Precalculus: An Investigation of Functions Ed. 1.5.
Retrieved from: http://www.opentextbookstore.com/precalc/