(a) Find the domains of f and f-1, (b) find the ranges of f and f-1 , and (c) graph f and f-1

Question

(a) Find the domains of f and f^-1, (b) find the ranges of f and f^-1 , (c) graph f and f^-1, and (d) show that the slopes of the graphs of f and f^-1 are reciprocals at the given points.

Functions                                          Points

Answer 1

(a)

(b)

Step 1:

The functions is and inverse function is .

The Domain of is equal to  the range of and the range of is equal to the domain of .

Find the domain and range of .

The cubic polynomial function is increasing function.

The domain of a function is all possible - values.

Domain of a polynomial function is always all real numbers.

Domain in interval notation .

Domain of is .

Step 2:

 Range of cubic function is all real numbers.

Range of in interval notation .

domain of is .

Range of is .

Solution:

(a) Domain of is and domain of is .

(b) Range of  is and range of is .

Answer 2

(c)

Step 1:

Graph the two functions  and inverse function is

Choose random values for and find the corresponding values for .

 

Step 2:

Graph:

Draw the coordinate plane.

Plot the points of and connect the curve.

Plot the points of and connect the curve.

.

Comments

Solution:

.

Answer 3

(d)

Step 1:

The functions are  and

First derivative is slope of the function.

Differentiate with respect to .

Apply power rule in derivatives .

The point is  .

Step 2:

Differentiate with respect to .

The point is

.

The slopes of the and are reciprocal at the points , .

Graph:

Solution:

The slopes of the and are reciprocal at the points , .