Q. 21 - 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, $\left(-2,0\right),\left(2,0\right),\left(0,3\right)$.
2. The domain of the given function is,$\left\{x|-4\le x\le 4\right\}$ and the range is, $\left\{y|0\le y\le 3\right\}$.
3. The given function $f$ is increasing on the open intervals $\left(-2,0\right)$ and $\left(2,4\right)$. $f$ is decreasing on the open intervals $\left(-4,-2\right)$ and $\left(0,2\right)$. $f$ is not constant at any point on the graph.
4. The function of the given graph is even.

Part a Step 1 . Given information

The given graph touches the $x$-axis at $\left(-2,0\right)$ and $\left(2,0\right)$ and touches the $y$-axis at $\left(0,3\right)$.

So, the intercepts are, $\left(-2,0\right),\left(2,0\right)$ and $\left(0,3\right)$.

Part b Step src="data:image/svg+xml;base64,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" role="math" localid="1647333217060" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/543fcb2c-82b1-45d2-a731-7e564d9366fe.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220315%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220315T090200Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=e8f00dc751b5b05bd34ebfe1585801c05e69d72bbdafac92ea71f8ab21fdfbd5" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/543fcb2c-82b1-45d2-a731-7e564d9366fe.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220315%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220315T090026Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=99516d9b0cfc6a9f9a94d2ad21579bd2ccb355e90ff82e317b87f59e882e1291" 1 . Find the domain and the range .

The domain of the given graph is, all the number between $-4$ and $4$, $\left\{x|-4\le x\le 4\right\}$ or $\left[-4,4\right]$.

The range of the given graph is, all the numbers between $0$ and $3$, $\left\{y|0\le y\le 3\right\}$ or $\left[0,3\right]$.

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

In the graph, the function $f$ is increasing from the point $\left(-2,0\right)$ to the point $\left(0,3\right)$ and from the point $\left(2,0\right)$ to the point $\left(4,2\right)$. So, it is increasing on the open intervals $\left(-2,0\right)$ and $\left(2,4\right)$.

In the graph, the function $f$ is decreasing from the point $\left(-4,2\right)$ to the point $\left(-2,0\right)$ and from the point $\left(0,3\right)$ to the point $\left(2,0\right)$. So, it is decreasing on the open intervals $\left(-4,-2\right)$ and $\left(0,2\right)$.

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

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

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

Therefore, the function of the given graph is even.