#### To determine

**a) To find:**

f is one to one.

#### Answer

f is one to one

#### Explanation

You can determine whether or not the given function is one to one by just looking at the graph with the help of Horizontal Line Test. Horizontal Line Test says that a function f is one to one if and only if any horizontal line intersects the graph of given function exactly at one point. In other words, the Horizontal Line Test can be stated as:

A function f is NOT one to one if and only if there exists a horizontal line which intersects the graph of given function at least at two point.

**Explanation by definition:** Let fx=f(y)⇒ x3=y3 ⇒x3-y3=0 ⇒x-yx2+xy+y2=0.

Here, x2+xy+y2 is a positive quantity for x>0,y>0 and for x<0, y<0. Therefore, x-y=0, i.e., x=y. Thus, f is one to one.

**Final statement:** Any horizontal line will intersect the given graph exactly once. Therefore, by using the Horizontal Line test, we can conclude that the given function is one to one.

#### To determine

**b)To find:**

The inverse of f.

#### Answer

f-1'8=112.

#### Explanation

By Theorem 7, f-1'a=1f'f-1a , where f'(f-1a)≠0.

**Step 1**: Substitute y=f(x).

That is,y=x3. By solving this equation for the value of x, we get x=y3.

**Step 2**: Interchange the positions of x and y, we get y=x3.

Thus,f-1x=x3.

**Calculation of**f-1'8: Now, f-18=83=2 and f'x=3x2. Also, f'f-18=f'2=12, which is non-zero.

**Final statement:** Therefore, by the above formula,

f-1'8=1f'f-18 =1f'2=112.

#### To determine

**c) To calculate:**

f
−1
(x) and its domain and range

#### Explanation

We have already calculated that f-1x=x3

Domain and range for f-1: For domain, look for values on the x-axis and for range, look for values on the y-axis. Domain and Range, both are the set of real numbers, (as f and f-1 are defined for all real numbers).

#### To determine

**d) To calculate:**

(
f
−1
)'(a)

#### Explanation

Since f-1x=x3,

f-1'x=13x23 ⇒ f-1'8=112, which agrees with the answer in Part (b)

#### To determine

**e) To sketch:**

The graphs of
f and
f
−1
on the same axes.

#### Explanation

The blue-colored graph represents f. The orange-colored graph represents f-1. The green-colored graph is the line y=x. You can see that the graph for f and f-1 are symmetrical about the line y=x.