#### To determine

**To explain:**

(a) How the inverse function f^{-1} is defined; and what is its domain and its range.

(b) If f is given, how will you find f^{-1}

(c) How will you find the graph of f^{-1} if the graph of f is given.

#### Answer

(a) Definition is given below:

(b) y=f-1x=x-2

(c) See the below (c) section.

#### Explanation

a) **Given**: f is a one-to-one function with domain A and range B.

**Definition of f-1:** The inverse function of f is a function with domain B and range A and is defined as follows:

f-1y=x⇔y=f(x), for all y in B.

**Note**: i) The definition says that if f maps x into y, then f-1 maps y to x.

ii) If *f* is not one to one, then f-1 cannot be uniquely defined.

b) **Steps to find f-1**:

**Step 1**: Substitute fx=y.

**Step 2**: Solve this equation to obtain the value of x in terms of y.

**Step 3:** To express f-1 as a function of x, interchange the positions of x and y.

The final equation we will get:y=f-1(x).

**Example:** Let us understand f-1 with the help of an example:

**Given**:fx=x+2,

Substitute y=f(x), i.e.,y=x+2.

Now solve the above equation to obtain the value of x in terms of y, i.e.,x=y-2.

Interchange the positions of x and y, i.e.,y=x-2. Hence,y=f-1x=x-2.

c) Suppose the graph for f is provided. Draw the graph for f-1, by the following steps:

**Step 1**: First draw the graph for Ix=x, where I is the identity map (the graph for the identity map is also called a diagonal) on the set of all real numbers.

**Step 2**: Take the mirror image of the graph for f about its diagonal.

The mirror image of the graph for f about the diagonal is the graph for f-1.

**Conclusion:**

(a) Defined

(b) y=f-1x=x-2

(c) See the (c) section.