Chapter 2, Problem 1RE

In Problem 1-3:

1. Determine the slope and y-intercept of each linear function.

2. Graph each function. Label the intercepts.

3. Determine the domain and the range of each function.

4. Determine the average rate of change of each function.

5. Determine whether the function is increasing, decreasing of constant.

$f\left(x\right)=2x-5$

To determine

The slope and y-intercept of the linear function $f\left(x\right)=2x-5$.

Explanation of Solution

Given information:

The linear function is $f\left(x\right)=2x-5$.

Explanation:

Let us consider the function,

$f\left(x\right)=2x-5$

Compare the given function $f\left(x\right)=2x-5$ with $f\left(x\right)=ax+b$ to obtain.

Hence, the slope $a=2$ and y- intercept $b=-5$

To determine

To graph: The function $f\left(x\right)=2x-5$ and label the intercept on the graph.

Explanation of Solution

Given information:

The linear function is $f\left(x\right)=2x-5$.

Graph:

Graph the function $f\left(x\right)=2x-5$ and label its intercepts.

Solve the equation $2x-5=0$ to evaluate the x-intercept of the given function.

$2x-5=0$

Now, add 5 on both sides below in value.

$2x=5$

Divide 2 on both sides below in value.

$x=\frac{5}{2}$

Therefore, the x-intercept is $x=\frac{5}{2}$

Then, the graph of the function $f\left(x\right)=2x-5$ using intercepts.

At first mark the x-intercept and y-intercepts on the axes and then join them with a straight edge

as shown in the following diagram.

To determine

The domain and the range of thefunction $f\left(x\right)=2x-5$.

Explanation of Solution

Given information:

The given function is $f\left(x\right)=2x-5$.

Explanation:

Find the domain and range of the given function $f\left(x\right)=2x-5$ which is negative.

Observe the above diagram, in which the function $f\left(x\right)=2x-5$ in a straight line.

The graph a along both sides of the x-axis without gaps, it means the domain of the given function is all real numbers.

Therefore the domain of the function $f\left(x\right)=2x-5$ $\left(-\infty ,\infty \right)$.

And also observed that, the graph is along both sides of the y-asix without gaps, it

Since, the range of the function $f\left(x\right)=2x-5$ is

Hence, the function f is decreasing. $\left(-\infty ,\infty \right)$

To determine

The average rate of change of the given function $f\left(x\right)=2x-5$.

Explanation of Solution

Given information:

The given function is $f\left(x\right)=2x-5$.

Explanation:

Calculate the average rate of the functions $f\left(x\right)=2x-5$

The average rate of the function $f\left(x\right)=2x-5$.

The average rate of a linear function is its slope: therefore the average rate of the function

$f\left(x\right)=2x-5$ is its slope which is equal to 2.

To determine

Whether the function is $f\left(x\right)=2x-5$ increasing, decreasing of constant.

Explanation of Solution

Given information:

The given function is $f\left(x\right)=2x-5$.

Explanation:

Determine whether the function is increasing, decreasing or constant.

Write the fact that, a linear function with positive slope is always an increasing function:

Apply the fact to check the behavior of the graph of the given function.

Therefore the given function $f\left(x\right)=2x-5$. is a linear function with positive slope 2.

Hence, the function $f\left(x\right)=2x-5$ is an increasing function.