Chapter 2.3, Problem 25AYU

In Problems 25-32, the graph of a function is given. Use the graph to find:

a. The intercepts, if any

b. The domain and range

c. The intervals on which the function is increasing, decreasing, or constant

d. Whether the function is even, odd, or neither

29.

To determine

To find: The following values using the given graph:

a. Intercepts ($x\text{and}y\text{intercepts}$).

Answer to Problem 25AYU

From the graph, concluding the following results:

a. Intercepts ($x\text{and}y\text{intercepts}$).

$\text{The}x\text{-intercept are} -\pi \text{and}0\text{the }y\text{-intercepts are }\pi \text{and} 0$.

Explanation of Solution

Given:

It is asked to find the intercepts ($x$ and $y$ if any), domain and range, increasing intervals, decreasing intervals, and constant intervals of the function using the graph. Also, check whether the function is even, odd or neither.

Graph:

Interpretation:

a. Intercepts ($x\text{and}y\text{intercepts}$): The points, if any, at which a graph crosses or touches the coordinate axes are called the intercepts.

The $x\text{-coordinate}$ of a point at which the graph crosses or touches the $x\text{-axis}$ is an $x\text{-intercept}$, and the $y\text{-coordinate}$ of a point at which the graph crosses or touches the $y\text{-axis}$ is an $y\text{-intercept}$.

The intercepts of the graph are the points $\left(-\pi ,0\right), \left(0,0\right)$ and $\left(0,\pi \right)$.

The $x\text{-intercepts}$ are $-\pi$ and 0; the $y\text{-intercepts}$ are $\pi$ and 0.

To determine

To find: The following values using the given graph:

b. The domain and range set of the function.

Answer to Problem 25AYU

From the graph, concluding the following results:

b. The domain and range set of the function.

The domain of $f$ is $\left\{x/-\pi \le x\le \pi \right\}$ or the interval $\left[-\pi ,\pi \right]$.

The range of $f$ is $\left\{x/-1\le x\le 1\right\}$ or the interval $\left[-1,1\right]$.

Explanation of Solution

Given:

It is asked to find the intercepts ($x$ and $y$ if any), domain and range, increasing intervals, decreasing intervals, and constant intervals of the function using the graph. Also, check whether the function is even, odd or neither.

Graph:

Interpretation:

b. The domain and range set of the function.

To determine the domain of $f$ notice that the points on the graph of $f$ have $x\text{-coordinates}$ between $-\pi$ to $\pi$, inclusive; and for each number $x$ between $-\pi$ to $\pi$, there is a point $\left(x,f\left(x\right)\right)$ on the graph. The domain of $f$ is $\left\{x/-\pi \le x\le \pi \right\}$ or the interval $\left[-\pi ,\pi \right]$.

The points on the graph all have $y\text{-coordinates}$ between $-1$ and 1 inclusive; and for each such number $y$, there is at least one number $x$ in the domain. The range of $f$ is $\left\{x/-1\le x\le 1\right\}$ or the interval $\left[-1,1\right]$.

To determine

To find: The following values using the given graph:

c. Increasing intervals, decreasing intervals and constant interval if any.

Answer to Problem 25AYU

From the graph, concluding the following results:

c. The function $f$ $\text{is increasing}\text{in the interval}\left[-\frac{\pi }{2}, \frac{\pi }{2}\right]$ and the function $f$ is $\text{decreasing in the intervals}\left[\pi , -\frac{\pi }{2}\right]\text{and}\left[\frac{\pi }{2},\pi \right]$. There is no constant interval in the given graph.

Explanation of Solution

Given:

It is asked to find the intercepts ($x$ and $y$ if any), domain and range, increasing intervals, decreasing intervals, and constant intervals of the function using the graph. Also, check whether the function is even, odd or neither.

Graph:

Interpretation:

c. Increasing intervals, decreasing intervals and constant interval if any.

It can be directly concluded from the graph that the curve is decreasing from $-\pi$ to $-\frac{\pi }{2}$, then increasing from $-\frac{\pi }{2}$ to $\frac{\pi }{2}$, and at last decreasing from $\frac{\pi }{2}$ to $\pi$.

Therefore, the function $f$ is increasing in the intervals $\left[-\frac{\pi }{2}, \frac{\pi }{2}\right]$ and the function $f$ is decreasing in the intervals $\left[\pi , -\frac{\pi }{2}\right]$ and $\left[\frac{\pi }{2},\pi \right]$. There is no constant interval in the given graph.

To determine

To find: The following values using the given graph:

d. Nature of the function (even, odd or neither).

Answer to Problem 25AYU

From the graph, concluding the following results:

d. The given function is symmetric with respect to the origin.

Explanation of Solution

Given:

It is asked to find the intercepts ($x$ and $y$ if any), domain and range, increasing intervals, decreasing intervals, and constant intervals of the function using the graph. Also, check whether the function is even, odd or neither.

Graph:

Interpretation:

d. Nature of the function (even, odd or neither).

By the theorem of test for symmetry, “A function is even if and only if its graph is symmetric with respect to the $y\text{-axis}$. A function is odd if and only if its graph is symmetric with respect to the origin”.

A graph is symmetric with respect to the origin if for every point $\left(a,b\right)$ on the graph, there is also a point $\left(-a,-b\right)$ on the graph.

It can be easily concluded from the graph and the above statement that the given function is symmetric with respect to the origin.