Q. 25 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 21–28, 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 it is increasing, decreasing, or constant

(d) Whether it is even, odd, or neither

Step-by-Step Solution

Answer

1. The intercepts are, $\left(-\mathrm{\pi },0\right),\left(\mathrm{\pi },0\right)$and$\left(0,0\right)$.
2. The domain of the function of the given graph is,$\left\{x|-\mathrm{\pi }\le \mathrm{x}\le \mathrm{\pi }\right\}$ and the range is, $\left\{y|-1\le y\le 1\right\}$.
3. In the graph, the function $f$ is increasing on the interval $\left(\frac{-\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$ and decreasing on the interval $\left(-\mathrm{\pi },\frac{-\mathrm{\pi }}{2}\right),\left(\frac{\mathrm{\pi }}{2},\mathrm{\pi }\right)$and the function is not constant at any points on the graph.
4. The function of the given graph is odd.

Part a Step 1 . Given information

The graph touches the $x$-axis at $\left(-\mathrm{\pi },0\right),\left(\mathrm{\pi },0\right)$ and $y$-axis at $\left(0,0\right)$.

So, the intercepts are,$\left(-\mathrm{\pi },0\right),\left(\mathrm{\pi },0\right)$ and $\left(0,0\right)$.

Part b Step 1 . Find the domain and the range

As $x$ varies from $-\mathrm{\pi }$ to $\mathrm{\pi }$, the domain of the function of the given graph is, $\left\{x|-\mathrm{\pi }\le \mathrm{x}\le \mathrm{\pi }\right\}$.

As $y$ varies from $-1$ to $1$, the range of the function of the given graph is, $\left\{y|-1\le y\le 1\right\}$.

Part c Step 1 . Determine the intervals on which it is increasing, decreasing, or constant

In the graph, the function $f$ is increasing from the point $\left(\frac{-\mathrm{\pi }}{2},-1\right)$ to the point $\left(\frac{\mathrm{\pi }}{2},1\right)$. So, it is increasing on the interval $\left(\frac{-\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$.

In the graph, the function $f$ is decreasing from the point $\left(-\mathrm{\pi },0\right)$ to the point $\left(\frac{-\mathrm{\pi }}{2},-1\right)$ and from the point $\left(\frac{\mathrm{\pi }}{2},1\right)$ to the point $\left(\mathrm{\pi },0\right)$. So, it is decreasing on the intervals $\left(-\mathrm{\pi },\frac{-\mathrm{\pi }}{2}\right)$ and $\left(\frac{\mathrm{\pi }}{2},\mathrm{\pi }\right)$.

The function $f$ is not constant at any points on the graph.

Part d Step 1 . Find whether the function of the given graph is even, odd, or neither

From the graph, we can see that for a point $\left(x,y\right)$ on the graph of $f$ $\left(-x,-y\right)$ is also on the graph. So, the given graph is symmetric with respect to the origin.

Therefore, the function is odd.