Chapter 5 Inverse Functions and Applications Section 5.3.

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

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.

1fg 1gf

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.

It

checks.

Therefore, f and g are inverses of each other.

36x

xgand6x3xf

36x

f))x(g(f)x)(gf(

x6)6x(636x

3

)6x3(g))x(f(g)x)(fg(

x3x3

36)6x3(

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,

12xxgand12x1

xf

)12x(f))x(g(f)x)(gf(

x24x1

12)12x(1

.)x(f

1)x(f 1

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)

5x)x(f 2

5xy 2

5yx 2

5xy2 5xy

5xxf 1

x

55x

5xfxff2

1

x

55x

5xfxff2

211

(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.

5xxfand5xxf 12

As of 2014, the sales tax rate in Tallahassee, Florida was one of the highest, at 7.5%. Source: www.sale-tax.com/Florida

a.If the function T(p)= 0.075p represents the sales tax inTallahassee, 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)

p075.0y y075.0p

075.0p

y

0750p

pT 1

.

(Contd.)

b. Evaluate and interpret T-1(18.75).

We know:

If the sales tax is $18.75, the price is $250.

25007507518

7518T 1

..

.

0750p

pT 1

.

(continued on the next slide)

(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.

p075.0p

075.0075.0p

T)p)(TT( 1

p075.0p075.0

)p075.0(T)p)(TT( 11

0750p

pTandp0750pT 1

..

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