Apply the Horizontal Line Test

Graph the function f (x) = 4x 2 + 4x + 1 using a

graphing calculator, and apply the horizontal line test to determine whether its inverse function exists. Write yes or no.

Apply the Horizontal Line Test

Graph the function f (x) = x 5 + x

3 – 1 using a graphing calculator, and apply the horizontal line test to determine whether its inverse function exists. Write yes or no.

Graph the function using a graphing

calculator, and apply the horizontal line test to determine whether its inverse function exists. Write yes or no.

Find Inverse Functions Algebraically

Determine whether f has an inverse function for

. If it does, find the inverse function

and state any restrictions on its domain.

The graph of f passes the horizontal line test. Therefore,

f is a one-one function and has an inverse function.

From the graph, you can see that f has domain

and range .

Now find f –1.

Find Inverse Functions Algebraically

Find Inverse Functions Algebraically

Original function

Replace f(x) with y.

Interchange x and y.

2xy – x = y Multiply each side by 2y – 1. Then apply the Distributive Property.

2xy – y = x Isolate the y-terms.

y(2x –1) = x Factor.

Find Inverse Functions Algebraically

Divide.

Find Inverse Functions Algebraically

Answer: f –1 exists;

From the graph, you can see that f –1 has domain

and range . The

domain and range of f is equal to the range and

domain of f –1, respectively. Therefore, it is not

necessary to restrict the domain of f –1.

Find Inverse Functions Algebraically

Determine whether f has an inverse function for . If it does, find the inverse function and state any restrictions on its domain.

Determine whether f has an inverse function for

. If it does, find the inverse function and

state any restrictions on its domain.

Verify Inverse Functions

Show that f [g (x)] = x and g [f (x)] = x.

Verify Inverse Functions

Because f [g (x)] = x and g [f (x)] = x, f (x) and g (x) are inverse functions. This is supported graphically because f (x) and g (x) appear to be reflections of each other in the line y = x.

Verify Inverse Functions

Answer:

Show that f (x) = x 2 – 2, x 0 and

are inverses of each other.

Find Inverse Functions Graphically

Use the graph of relation A to sketch the graph of its inverse.

Answer:

Find Inverse Functions Graphically

Graph the line y = x. Locate a few points on the graph of f (x). Reflect these points in y = x. Then connect them with a smooth curve that mirrors the curvature of f (x) in line y = x.

Use the graph of the function to graph its inverse function.

Use an Inverse Function

MANUFACTURING The fixed costs for manufacturing one type of stereo system are $96,000 with variable cost of $80 per unit. The total cost f (x) of making x stereos is given by f (x) = 96,000 + 80x. Find f –1(x) and explain what f –1(x) and x represent in the inverse function.