#### 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)

⇒ x-2=y-2 . After squaring both sides, we get x-2=y-2⇒ 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'a=4.

#### Explanation

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

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

That is, y=x-2 ⇒x=y2+2.

**Step 2**: Interchange the positions of x and y, we get y=x2+2.

Thus f-1x=x2+2.

**Calculation of** (f-1)'(a): Now, f-12=6 and the derivative of f is f'x=12x-2. So,

f'f-1a=f'6=14, which is non-zero.

**Final statement:** Therefore, f-1'a=4.

#### To determine

**c) To calculate:**

f
−1
(x) and its domain and range for f(x)=
x−2

#### Explanation

We have already calculated f-1x=x2+2.

Domain: Since square root function is defined for non-negative real numbers, given function is defined for x≥2.

Range: The set of all real numbers.

#### To determine

**c) To calculate:**

(
f
−1
)'(a) for f(x)=
x−2

#### Explanation

f-1x=x2+2. Therefore,f-1'x=2x ⇒(f-1)'2=4, which is the same as we calculated in Part (b) using Theorem 7.

#### To determine

**e) To sketch:**

The graphs of
f and
f
−1
on the same axes for f(x)=
x−2

#### Explanation

The orange colored graph represents f-1, blue colored graph represents f. You can observe that the graph of f and f-1 are symmetrical about the line y=x.