Chapter 1, Problem 2RE

(a)

To determine

To state: The value of $g\left(2\right)$.

Answer to Problem 2RE

Solution:

The value of $g\left(2\right)$ is 3.

Explanation of Solution

Let $y=g\left(x\right)$ be the function.

From the given graph, it is noticed that when $x=2$, then the curve approaches to 3.

That is, $g\left(2\right)=3$.

(b)

To determine

To explain: Why the function g is one-to-one.

Explanation of Solution

The function is one-to-one if it satisfies the horizontal line test.

Horizontal line test states that, if the horizontal line intersects the curve atmost once then the function g is one-to-one.

From the given graph, it is noticed that no horizontal line intersects the graph more than once. So, it satisfies the horizontal line test.

Hence, the function f is one-to-one.

(c)

To determine

To estimate: The values of $g^{-1}\left(2\right)$.

Answer to Problem 2RE

Solution:

The values of $g^{-1}\left(2\right)$ is approximately equal to 2.5.

Explanation of Solution

Let $y=g\left(x\right)$ be the function, then $g^{-1}\left(y\right)=x$.

From the given graph, it is noticed that the value of $g\left(x\right)=2$ when $x=2.5$.

Then, it can be implied that $g^{-1}\left(2\right)\approx 2.5$

When $y=2$, then $g^{-1}\left(2\right)\approx 2.5$.

(d)

To determine

To estimate: The domain of $g^{-1}$.

Answer to Problem 2RE

Solution:

The domain of $g^{-1}$ is $\left[-1,3.5\right]$.

Explanation of Solution

It is known that the domain of $g^{-1}$ is the range of $g$ and the range of $g^{-1}$ is the domain of $g$.

From the given graph, it is observed that the range of g is $\left[-1,3.5\right]$.

Therefore, the domain of $g^{-1}$ is $\left[-1,3.5\right]$.

(e)

To determine

To sketch: The graph of $g^{-1}$.

Explanation of Solution

Graph:

The graph of $g^{-1}$ is obtained by taking the reflection of $g$ about the line $y=x$. Sketch the graph of $g^{-1}$ as shown below in Figure 1.

From Figure1, it is observed that g is the reflection of $g^{-1}$ about the line $y=x$.