Q. 22 - 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, $\left(-1,0\right),\left(1,0\right)$ and $\left(0,2\right)$.
2. The domain of the function for the given graph is, $\left\{x|-3\le x\le 3\right\}$ and the range is, $\left\{y|0\le y\le 3\right\}$.
3. The function $f$ is increasing on the intervals $\left(-1,0\right)$ and $\left(1,3\right)$ and decreasing on the intervals $\left(-3,-1\right)$ and $\left(0,1\right)$ and the graph is not constant anywhere in the domain.
4. Th function of graph is even.

Part a Step 1 . Given information

The graph touches the $x$-axis at $\left(-1,0\right)$ and $\left(1,0\right)$ and touches the $y$-axis at $\left(0,2\right)$.

So, the intercepts are, $\left(-1,0\right),\left(1,0\right)$ and $\left(0,2\right)$.

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

In the graph, we can see $x$ varies from $-3$ to $3$. Hence the domain is, $\left\{x|-3\le x\le 3\right\}$.

We can see $y$ varies from $0$ to $3$. Hence the range is, $\left\{y|0\le y\le 3\right\}$.

Part c Step 3 . Determine the intervals on which it is increasing, decreasing, or constant

Consider the graph, it is rising (increasing) from the point $\left(-1,0\right)$ to $\left(0,2\right)$.

Also, the graph is rising from the point $\left(1,0\right)$ to $\left(3,3\right)$.

Hence, the function $f$ is increasing on the intervals $\left(-1,0\right)$ and $\left(1,3\right)$.

The given graph is decreasing from the point $\left(-3,3\right)$ to $\left(-1,0\right)$.

Also, the graph is decreasing from the point $\left(0,2\right)$ to $\left(1,0\right)$.

Hence, the function $f$ is decreasing on the intervals $\left(-3,-1\right)$ and $\left(0,1\right)$.

Part d Step 4 . Whether it is even, odd, or neither

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