Chapter 2, Problem 1RE

(a)

To determine

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

Answer to Problem 1RE

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

Explanation of Solution

Definition used:

Linear Function:

A linear function is a function of the form $f\left(x\right)=mx+b$. The graph of a linear function is a line with slope m and y-intercept b. Its domain is the set of all real numbers.

Calculation:

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

Compare the function $f\left(x\right)=2x-5$ with the linear function $f\left(x\right)=mx+b$.

Here, $m=2\text{ and }b=-5$.

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

(b)

To determine

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

Explanation of Solution

From part (a), the slope and y-intercept of $f\left(x\right)=2x-5$ are 2 and $-5$, respectively.

Evaluate the x-intercept of $f\left(x\right)=2x-5$ as follows.

$\begin{array}{l}f\left(x\right)=0\\ 2x-5=0\\ 2x=5\\ x=\frac{5}{2}\end{array}$

Thus, the x-intercept is $\left(\frac{5}{2},0\right)$.

To sketch the graph of $f\left(x\right)=2x-5$, plot the y-intercept $\left(0,-5\right)$ and using the slope find an additional point by moving 1 unit right and 2 units up.

The graph of the function $f\left(x\right)=2x-5$ is shown below in Figure 1.

From Figure 1, observe that the graph of $f\left(x\right)=2x-5$ is a straight line with y-intercept $\left(0,-5\right)$ and x-intercept $\left(\frac{5}{2},0\right)$.

(c)

To determine

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

Answer to Problem 1RE

The domain of $f\left(x\right)=2x-5$ is $\left(-\infty ,\infty \right)$.

The range of $f\left(x\right)=2x-5$ is $\left(-\infty ,\infty \right)$.

Explanation of Solution

Definitions used:

Domain:

The domain is the set of all x values for which the function is defined.

Range:

Range is the set of all output values of a function.

Calculation:

The function $f\left(x\right)=2x-5$ is defined for all the values of x.

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

From the graph of $f\left(x\right)=2x-5$ in part (b), it is observed that the range is $\left(-\infty ,\infty \right)$.

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

(d)

To determine

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

Answer to Problem 1RE

The average rate of change is 2.

Explanation of Solution

Theorem used:

Average rate of change of a linear function:

Linear functions have a constant average rate of change. That is, the average rate of change of a linear function $f\left(x\right)=mx+b$ is $\frac{\Delta y}{\Delta x}=m$.

Calculation:

From part (a), the slope of $f\left(x\right)=2x-5$ is 2.

By the above mentioned theorem, the average rate of change is $\frac{\Delta y}{\Delta x}=2$.

Therefore, the average rate of change is 2.

(e)

To determine

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

Answer to Problem 1RE

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

Explanation of Solution

Theorem used:

Increasing, decreasing and constant Linear Functions:

A linear function $f\left(x\right)=mx+b$ is increasing over its domain if its slope, m is positive. It is decreasing over its domain, if its slope is negative. It is constant over its domain, if its slope m is zero.

Calculation:

From part (a), the slope of $f\left(x\right)=2x-5$ is 2, which is positive.

So, by the above theorem, the function $f\left(x\right)=2x-5$ is increasing.

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