Chapter 7, Problem 7CR

(a)

To determine

To graph: The function $y=x^3$ and label three points on it. Also, find the inverse of the function and draw its graph.

Explanation of Solution

Given information:

The function $y=x^3$.

Graph:

For labelling, take three values of $x$ and find the values of $y$ using the function $y=x^3$.

$\begin{array}{ccc}x& y=x^3& \left(x,y\right)\\ -1& -1& \left(-1,-1\right)\\ -0.75& 0.421875& \left(-0.75,0.421875\right)\\ 0& 0& (0,0)\\ 0.75& 0.421875& (0.75,0.421875)\\ 1& 1& (1,1)\end{array}$

The graph of the function $y=x^3$ is as follows:

To find the inverse function $y=x^3$, first interchange the variables.

$x=y^3$

Now solve $x=y^3$ for $y$.

Take the cube root of each side.

$\Rightarrow {\left(x\right)}^{\frac{1}{3}}={\left(y^3\right)}^{\frac{1}{3}}$

$\Rightarrow y={\left(x\right)}^{\frac{1}{3}}$

The inverse function of $y=x^3$ is $y={\left(x\right)}^{\frac{1}{3}}$.

The graph of the inverse function is a reflection about the line $y=x$.

Therefore, the graph of the inverse function $y={\left(x\right)}^{\frac{1}{3}}$ is as follows:

Interpretation:

The graph of $y=x^3$ and the graph of the inverse function are symmetric around the origin.

(b)

To determine

To graph: The function $y=e^x$ and label three points on it. Also, find the inverse of the function and draw its graph.

Explanation of Solution

Given information:

The function, $y=e^x$

Graph:

For labelling, take three values of $x$ and find the values of $y$ using the function $y=e^x$.

$\begin{array}{ccc}x& y=e^x& \left(x,y\right)\\ -2& 0.135& \left(-2,0.135\right)\\ -1& 0.368& \left(-1,0.368\right)\\ 0& 1& \left(0,1\right)\end{array}$

The graph of the function $y=e^x$ is as follows:

To find the inverse function $y=e^x$, first interchange the variables.

$x=e^y$

Now solve $x=e^y$ for $y$.

Take log in both side.

$\Rightarrow \log \left(x\right)=\log \left(e^y\right)$

$\Rightarrow y=\log x$

The inverse function of $y=e^x$ is $y=\log x$.

As the graph of the inverse function is reflection about the line $y=x$

Therefore, the graph of the inverse function $y={\left(x\right)}^{\frac{1}{3}}$ is as follows:

Interpretation:

The graph of $y=e^x$ is the reflection graph of the inverse function $y=\log x$ about the line $y=x$.

(c)

To determine

To graph: The function $y=\sin x,-\frac{\pi }{2}\le x\le \frac{\pi }{2}$ and label three points on it. Also, find the inverse of the function and draw its graph.

Explanation of Solution

Given information:

The function, $y=\sin x,-\frac{\pi }{2}\le x\le \frac{\pi }{2}$

Graph:

For labelling, take three values of $x$ and find the values of $y$ using the function $y=\sin x,-\frac{\pi }{2}\le x\le \frac{\pi }{2}$.

$\begin{array}{ccc}x& y=\sin x& (x,y)\\ -1& -0.841& (-1,-0.841)\\ -0.5& -0.479& (-0.5,-0.479)\\ 0& 0& (0,0)\\ 0.5& -.479& (0.5,0.479)\\ 1& 0.841& (1,0.841)\end{array}$

The graph of the function $y=\sin x,-\frac{\pi }{2}\le x\le \frac{\pi }{2}$ is as follows:

To find the inverse function $y=\sin x$, first interchange the variables.

$x=\sin y$

Now solve $x=\sin y$ for $y$.

$\Rightarrow y={\sin }^{-1}\left(x\right)$

The inverse function of $y=\sin x$ is $y={\sin }^{-1}\left(x\right)$.

As the graph of the inverse function is a reflection about the line $y=x$

Therefore, the graph of the inverse function $y={\left(x\right)}^{\frac{1}{3}}$ is as follows:

Interpretation:

The graph of $y=\sin x$ is the reflection graph of the inverse function $y={\sin }^{-1}\left(x\right)$ about the line $y=x$.

(d)

To determine

To graph: The function $y=\cos x,0\le x\le \pi$ and label three points on it. Also, find the inverse of the function and draw its graph.

Explanation of Solution

Given information:

The function, $y=\cos x,0\le x\le \pi$

Graph:

For labelling, take three values of $x$ and find the values of $y$ using the function $y=\cos x,0\le x\le \pi$.

$\begin{array}{ccc}x& y=\cos x& (x,y)\\ 0& 1& (0,1)\\ 1& 0.54& (1,0.54)\\ 1.57& 0& (1.57,0)\\ 2.14& -0.54& (2.14,-0.54)\\ 3.14& -1& (3.14,-1)\end{array}$

The graph of the function $y=\cos x,0\le x\le \pi$ is as follows:

To find the inverse function $y=\cos x$, first interchange the variables.

$x=\cos y$

Now solve $x=\cos y$ for $y$.

$\Rightarrow y={\cos }^{-1}\left(x\right)$

The inverse function of $y=\cos x$ is $y={\cos }^{-1}\left(x\right)$.

As the graph of the inverse function is a reflection about the line $y=x$

Therefore, the graph of the inverse function $y=\cos x$ is as follows:

Interpretation:

The graph of $y=\cos x$ is a sinusoidal curve around $x$-axis, whereas the inverse function $y={\cos }^{-1}\left(x\right)$ is a sinusoidal curve around $y$-axis