Chapter 1.4, Problem 36E

a.

To determine

Find the difference between $y=f(g(x))$ and graph of the sine function.

Answer to Problem 36E

Hence, we will get the blurred image in the graph.

Explanation of Solution

Given information:

This exercise explores the effect of the inner function $g$ on a composite function $y=f(g(x))$ .

Graph the function $y=\sin \left(\sqrt{x}\right)$ using the viewing rectangle $[0,400]by[-1.5,1.5]$ . How does this graph differ from the graph of the sine function?

Calculation:

The graph of the function $y=\sin \left(\sqrt{x}\right)$ using the viewing rectangle $[0,400]\times [-1.5,1.5]$ is shown below:

  

The graph of the function $y=\sin (x)$ using the viewing rectangle $[0,400]\times [-1.5,1.5]$ is shown below:

  

We have $\sqrt{x}\le x$

That is, the values of the angle of the function $y=\sin (x)$ are greater than the values of the angle of the function $y=\sin \left(\sqrt{x}\right)$

So, we will get the more number of maxima, and minima for the graph of $y=\sin (x)$ when compared with the graph of $y=\sin \left(\sqrt{x}\right)$

Hence, we will get the blurred image in the graph of $y=\sin (x)$ by joining these maxima and minima, when compared with graph the graph of $y=\sin \left(\sqrt{x}\right)$

b.

To determine

Find the difference between $y=f(g(x))$ and graph of the sine function.

Answer to Problem 36E

Hence, we will get the blurred image in the graph.

Explanation of Solution

Given information:

This exercise explores the effect of the inner function $g$ on a composite function $y=f(g(x))$ .

Graph the function $y=\sin \left(x^2\right)$ using the viewing rectangle $[-5,5]by[-1.5,1.5]$ . How does this graph differ from the graph of the sine function?

Calculation:

The graph of the function $y=\sin (x^2)$ using the viewing rectangle $[-5,5]\times [-1.5,1.5]$ is shown below:

  

The graph of the function $y=\sin (x)$ using the viewing rectangle $[-5,5]\times [-1.5,1.5]$ is shown below:

  

We have $x\le x^2$

That is, the values of the angle of the function $y=\sin (x^2)$ are greater than the values of the angle of the function $y=\sin \left(x\right)$

So, we will get the more number of maxima, and minima for the graph of $y=\sin (x^2)$ when compared with the graph of $y=\sin \left(x\right)$

So that, we will get the blurred image in the graph of $y=\sin (x^2)$ by joining these maxima and minima, when compared with graph the graph of $y=\sin \left(x\right)$