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Problem 73

Think About It Sketch the graphs of

$$f(x)=\sin x, \quad g(x)=|\sin x|, \quad \text { and } \quad h(x)=\sin (|x|)$$

In general, how are the graphs of $|f(x)|$ and $f(|x|)$ related to the graph of $f ?$

Answer

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## Discussion

## Video Transcript

All right, let's explore this idea of f of the absolute value of X. I'm going to make a table of values and think this through. So if X is one, then the Y coordinate is just f of one. But if X's negative one, then we're going to take the absolute value of negative one, and then we'll find the y coordinate. So we're just going to get off of one again. If X is too, will get f of to. But if X is negative two, we're going to take the absolute value of that before we apply it to F. So we're going to get negative to again. So the right side of the equation, the right side of the function, whatever it looked like, it doesn't really matter. What's going to happen is we're going to see that reflected over on the left side, so f of the absolute value of X will have. Why access, symmetry, whatever we saw on the right is going to be reflected over to the left. So let's take a look at wide calls. The sign of the absolute value. Becks. First, let's take a look at why equal sign of X. The right side of it goes upto one down to negative one has a period of two pi. So this is what we normally see for a sign graph, and it would continue on to the right. What we're going to do is reflect that across the y axis, we'll see the mirror image on the left. Same idea for y equals the square root of the absolute value of X. Typically the square root of X looks like this. You have 00 You have 11 you have four to you have this shape. Typically, don't have anything over on the left because you can't take the at the square root of a negative and get a real output. However, before we take the square root, we're going to be taking the absolute value that negative. So we're going to see the reflection of that across the other side. So here's a graph

## Recommended Questions

(a) How is the graph of $ y = f (\mid x \mid) $ related to the graph of $ f $.

(b) Sketch the graph of $ y = \sin \mid x \mid $.

(c) Sketch the graph of $ y = \sqrt{\mid x \mid} $.

(a) Graph the function $ f(x) = \sin (\sin^{-1} x) $ and explain the appearance of the graph.

(b) Graph the function $ g(x) = \sin^{-1} (\sin x) $. How do you explain the appearance of this graph?

Graph each of the following.

$$f(x)=x \sin x$$

How does the graph of $y=\sin x$ compare with the graph of $y=\cos x ?$ Explain how you could horizontally translate the graph of $y=\sin x$ to obtain $y=\cos x .$

Graph $f(x)=\sin ^{-1} x$ together with its first two derivatives. Comment on the behavior of $f$ and the shape of its graph in relation to the signs and values of $f^{\prime}$ and $f^{\prime \prime}.$

THINK ABOUT IT Consider the functions given by

$f(x)= \sin\ x$ and $f^{-1}(x)= \arcsin\ x$.

(a) Use a graphing utility to graph the composite functions $f \circ f^{-1}$ and $f^{-1} \circ f$.

(b) Explain why the graphs in part (a) are not the graph of the line $y=x$. Why do the graphs of $f \circ f^{-1}$ and $f^{-1} \circ f$ differ?

Graph the function.

$$

f(x)=-\sin x

$$

Graph the function

\begin{equation}f(x)=\sin x \sin (x+2)-\sin ^{2}(x+1).\end{equation}

What does the graph do? Why does the function behave this way? Give reasons for your answers.

The functions $f(x)=\sin \left(\sin ^{-1} x\right) \quad$ and $\quad g(x)=\sin ^{-1}(\sin x)$ both simplify to just $x$ for suitable values of $x$ . But these functions are not the same for all $x$ . Graph both $f$ and $g$ to show how the functions differ. (Think carefully about the domain and range of $\sin ^{-1} $.)

Describe a relationship between the graphs of $y=\sin x$ and $y=\cos x$