Problem 9

Question

Draw the graphs of $$ f(x)=x+\cos x $$

Step-by-Step Solution

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Answer

The graph of $f(x) = x + \cos(x)$ is a wave that increases linearly in amplitude due to the linear $x$ term in the function.

1Step 1: Understand the function composition

The function $f(x) = x + \cos(x)$ is composed of a linear function $x$ and a cosine function $\cos(x)$. The linear function $x$ is a straight line passing through the origin with a slope of 1. The cosine function $\cos(x)$ oscillates between -1 and 1, and has a period of $2\pi$ (it repeats itself every $2\pi$ units).

2Step 2: Draw the individual graphs

First, sketch the individual graphs of $y = x$ and $y = \cos(x)$ on the same set of axes. The graph of $y = x$ is a straight line passing through the origin, and the graph of $y = \cos(x)$ is a wave that oscillates between -1 and 1.

3Step 3: Combine the graphs

Now, draw the graph of $f(x) = x + \cos(x)$. At each point $x$, the value of $f(x)$ is the sum of the corresponding values of $x$ and $\cos(x)$. Because the cosine function is cyclical, the graph of $f(x)$ will be a wave that increases linearly.

Problem 9

Other exercises in this chapter

Problem 8

Draw the graphs of $$ f(x)=\sin \left(x+\frac{\pi}{6}\right) $$

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Problem 9

Draw the graphs of $$ f(x)=\log _{2}(\sin x) $$

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Problem 9

Draw the graphs of $$ f(x)=\sin (x-1) $$

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Problem 10

Draw the graphs of $$ f(x)=\log _{2}(\cos x) $$

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