Calculate f -1 (x) and state the domain and range of f -1 .

Question

(a) Show that f is one-to-one.

(b) To find (f ^-1) ' (a).

(c) Calculate f ^-1 (x) and state the domain and range of f ^-1 .

(d) Calculate (f ^-1) ' (a) from the formula in part (c) and check that it agrees with the result of part (b).

(e) Sketch the graphs of f and f ^-1 on the same axes.

f (x) = x^3,              a = 8

Answer 1

Step 1 :

(a)

The function is .

A function is said to be one to one if any two elements in the domain are correspond to two different elements in the range.

If and are two different inputs of a function , then is said to be one to one provided  .

The domain of the function is all real number set.

Take and in the domain set.

.

Since , the function   is said to be one-to-one function.

Step 2 :

(c)

The function is .

Let .

To find the inverse of , replace x with y and y with x.

Solve for y.

.

The inverse of the function is .

The domain of a function is all values of x, those makes the function mathematically correct.

Since there shouldnot be any negative numbers in the cube root.

So, the domain of the above function is all non negative real numbers.

Domain of is .

Range set is the corresponding values of the function for different values of x.

Since for all non negative real numbers of x, the function is greater than equals to zero.

The range of the function is always greater than or equal to zero.

Range of is : .

Comments

Contd.....

Step 3:

(b)

The inverse function is .

 Differentiate the function with respect to x.

Apply power rule of derivatives : .

Find at .

.

Step 4 :

(d)

.

At , .

 

.

Step 5 :

(e)

The graph of and is :

 

 Solution :

(a) The function   is said to be one-to-one function.

(b) .

(c) The inverse function is , and its domain and range is and
.

(d) .

(e)The graph is :