Chapter 2, Problem 5CT

Consider the graph of the function $f$ below.

Find the domain and the range of $f.$

List the intercepts.

Find $f(1).$

For what value(s) of $x$ does $f(x)=-3?$

Solve $f(x)<0.$

To determine

The domain and range of the function $f$ where the graph of function is given by:

Answer to Problem 5CT

Solution:

The domain of the function $f$ is $[-5,5]$.

The range of the function $f$ is $[-3,3]$.

Explanation of Solution

Given Information:

The graph of function is:

Explanation:

Domain is set of $x$ values where the function is defined.

Here, function is defined for the values $-5\text{to}5$, inclusive.

Therefore, domain of the function $f$ is $[-5,5]$.

Range is the set of $y$ values corresponding to $x$ values.

Here, minimum $y$ value is $-3$.

Maximum $y$ value is 3.

Therefore, the range of the function $f$ is $[-3,3]$.

To determine

The intercepts of the function $f$ from the graph

Answer to Problem 5CT

Solution:

The $x$ intercepts of the function $f$ are $x=-2,x=2$, and the $y$ intercept is $y=2$.

Explanation of Solution

Given Information:

The graph of function is:

Explanation:

$x$ intercept is the point where the graph crosses $x-\text{axis}$.

Therefore, the $x$ intercepts are $x=-2,x=2$.

$y$ intercept is the point where the graph crosses $y-\text{axis}$.

Therefore, the $y$ intercept is $y=2$.

To determine

The value of $f(1)$ from graph of the function $f$ from the graph

Answer to Problem 5CT

Solution:

The value of $f(1)=3$.

Explanation of Solution

Given Information:

The graph of function is:

Explanation:

$f(1)$ is the $y$ value for which $x$ value is 1.

From above graph, at $x=1$, the point on the graph is $\left(1,3\right)$.

Thus, the value of $f(1)=3$.

To determine

For what values of $x$, $f(x)=-3$ from the graph

Answer to Problem 5CT

Solution:

For $x=-5\text{and}x=3$, $f(x)=-3$.

Explanation of Solution

Given Information:

The graph of function is:

Explanation:

From the graph, the $y$ value is $-3$ for $x=-5\text{and}x=3$.

Therefore, for $x=-5\text{and}x=3$, $f(x)=-3$.

To determine

The values of $x$ for which $f(x)<0$, where the graph of the function is shown in part a).

Answer to Problem 5CT

Solution:

The function $f(x)<0$ for $\left\{x|x\in \left[-5,-2\right)\cup \left(2,5\right]\right\}$.

Explanation of Solution

Given Information:

The graph of function is:

Explanation:

The function $f(x)<0$ means finding those values of $x$ for which $y$ values are negative.

From the graph, the $y$ values are negative for the intervals $\left[-5,-2\right)$ and $\left(2,5\right]$.

Therefore, $f(x)<0$ for $\left\{x|x\in \left[-5,-2\right)\cup \left(2,5\right]\right\}$.