1. Chapter 5 Inverse Functions and Applications Section 5.3

2. Section 5.3 Determining the Inverse Using Composition of Functions
Using Composition to Verify Inverse Functions
Applications

3. Two functions f and g are inverses of each other if and only if:
Inverse Functions
Two functions f and g are inverses of each other if and only if:
(f ◦ g)(x) = f(g(x)) = x for every x in the domain of g
and
(g ◦ f)(x) = g(f(x)) = x for every x in the domain of f.
So, and
The domain of f is equivalent to the range of g and vice versa.

4. If g(x) is the inverse of f(x), then (f ◦ g)(x) = x.
Determine whether the given functions f and g are inverses of each other.
If g(x) is the inverse of f(x), then (f ◦ g)(x) = x.
It checks.
If f(x) is the inverse of g(x), then (g ◦ f)(x) = x.
Therefore, f and g are inverses of each other.

5. Determine whether the given functions f and g are inverses of each other.
Let us find f ◦ g.
Since f ◦ g  x, there is no need to find g ◦ f.
Functions f and g are not inverses of each other.
Note:
Recall from Section 5.1 that f-1(x) is the notation for the inverse function and it does not mean the reciprocal of f(x). That is,

6. Find the inverse function of for x  0
Find the inverse function of for x  0. Verify algebraically and graphically that f and f-1(x) are inverses of each other.
The inverse is:
Verifying:
(continued on the next slide)

7. The graphs of f and f-1(x) for the specified domain are shown next.
(Contd.)
The graphs of f and f-1(x) for the specified domain are shown next.
Observe that the graphs of the functions are symmetric with respect to y = x.

8. If the function T(p)= 0.075p represents the sales tax in
As of 2014, the sales tax rate in Tallahassee, Florida was one of the highest, at 7.5%. Source:
If the function T(p)= 0.075p represents the sales tax in
Tallahassee, Florida for price p, determine T-1(p) and explain its meaning in the context of this problem.
The inverse is:
The inverse gives the price of an item or service if p dollars are paid as sales tax.
(continued on the next slide)

9. b. Evaluate and interpret T-1(18.75). We know:
(Contd.)
b. Evaluate and interpret T-1(18.75).
We know:
If the sales tax is $18.75, the price is $250.
(continued on the next slide)

10. c. Show that T and T-1 are inverses of each other.
(Contd.)
c. Show that T and T-1 are inverses of each other.
Since the result is p for both compositions, T and T-1 are inverses of each other.

11. Using your textbook, practice the problems assigned by your instructor to review the concepts from Section 5.3.