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

Short 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.

Step-by-step solution

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.