Q. 26 - 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, $\left(-\frac{\mathrm{\pi }}{2},0\right),\left(\frac{\mathrm{\pi }}{2},0\right)$ and $\left(0,1\right)$.
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 $\left(-\mathrm{\pi },0\right)$ and decreasing on the interval $\left(0,\mathrm{\pi }\right)$ and is not constant anywhere in its domain.
4. The function of the given graph is even.

Part a Step 1 . Given information

The given graph touches the $x$-axis at $\left(\frac{-\mathrm{\pi }}{2},0\right),\left(\frac{\mathrm{\pi }}{2},0\right)$ and $y$-axis at $\left(0,1\right)$.

So, the intercepts are, $\left(\frac{-\mathrm{\pi }}{2},0\right),\left(\frac{\mathrm{\pi }}{2},0\right)$ and $\left(0,1\right)$.

Part b Step 1 . Find the domain and the range.

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

The range is, $\left\{y|-1\le y\le 1\right\}$ as $y$ varies from $-1$ to $1$.

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

The graph is increasing from the point $\left(-\mathrm{\pi },-1\right)$ to the point $\left(0,1\right)$. Hence, the function $f$ is increasing on the interval $\left(-\mathrm{\pi },0\right)$.

The graph is decreasing from the point $\left(0,1\right)$ to the point $\left(\mathrm{\pi },-1\right)$.

Hence, the function $f$ is decreasing on the interval $\left(0,\mathrm{\pi }\right)$.

The graph is not constant anywhere in its domain.

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

The function of the given graph is even. Because, the graph is symmetric with respect to the $y$-axis.