Chapter 5 Inverse Functions and Applications Section 5.3

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

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.

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.

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,

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)

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.

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: www.sale-tax.com/Florida 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)

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)

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.

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