Chapter 5.3, Problem 101E

a.

To determine

To find: Domain of the given function.

Answer to Problem 101E

The domain of $f(x)$ is all real values except $0$ .

Explanation of Solution

Given information:

The following function:

  $f(x)=\cos \frac{1}{x}$

The following graph:

  

The range in which all possible inputs of a function fall, is called the domain of a function.

According to the graph, $f(x)$ does not has fixed value at $x=0$ . Therefore, the domain of the function will be:

All real values except $0$ .

b.

To determine

To identify: Any symmetry and asymptotes of the graph.

Answer to Problem 101E

The graph has y-axis symmetry.

The graph has a horizontal asymptote at $y=1$ as $x\to \infty$

Explanation of Solution

Given information:

The following function:

  $f(x)=\cos \frac{1}{x}$

The following graph:

  

The graph on the right side of y-axis appears to be a mirror image of the graph on the left side of the y-axis.

Hence the graph has y-axis symmetry.

An asymptote is a line representing a value towards which a function may approach but does not approach.

The graph has a horizontal asymptote at $y=1$ as $x\to \infty$ .

c.

To determine

To describe: Behavior of the function at $x\to 0$ .

Answer to Problem 101E

As $x\to 0$ , the graph oscillates between $-1$ and $1$ .

Explanation of Solution

Given information:

The following function:

  $f(x)=\cos \frac{1}{x}$

The following graph:

  

According to the graph, as $x\to 0$ , the function does not has a fixed value. It oscillates between $-1$ and $1$ . Hence the function is undefined as $x\to 0$ .

d.

To determine

To find: The solution of the given equation in the given interval.

Answer to Problem 101E

The equation has infinite number of solutions in the interval $[-1,1]$ and

  $x=\frac{2}{\pi +2n\pi }$

Explanation of Solution

Given information:

The following equation:

  $\cos \frac{1}{x}=0$

The following interval:

  $[-1,1]$

The following graph:

  

According to the graph, as $x\to 0$ , the function does not has a fixed value. It oscillates between $-1$ and $1$ . Therefore, the equation has infinite number of solutions in the interval $[-1,1]$ .

The solutions of the equation are calculated as:

  $\begin{array}{l}\cos \frac{1}{x}=0\text{ [Given]}\\ \frac{1}{x}=\arccos (0)\text{ [Inverse trigonometric function]}\\ \frac{1}{x}=\frac{\pi }{2}+n\pi \text{ [Value of }\arccos (0)]\\ \frac{1}{x}=\frac{\pi +2n\pi }{2}\text{ [Add fractions]}\\ x=\frac{2}{\pi +2n\pi }\text{ [Take reciprocal on both sides]}\end{array}$

e.

To determine

To find: The greatest solution of the given equation.

Answer to Problem 101E

The greatest solution is $0.6366$ .

Explanation of Solution

Given information:

The following equation:

  $\cos \frac{1}{x}=0$

The following interval:

  $[-1,1]$

The following graph:

  

Graph:

Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expression $\cos \frac{1}{x}=0$ which is required to graph.

Step 4: Press GRAPH button to graph the function.

The graph is obtained as:

  

Interpretation:

The maximum value of $x=0.6366$ .