Chapter 4, Problem 1PRE

For Exercises 1-14,
a. Write the domain.
b. Write the range.
c. Find the x-intercept(s).
d. Find the y-intercept.
e. Determine the asymptotes if applicable.
f. Determine the intervals over which the function is increasing.
g. Determine the intervals over which the function is decreasing.
h. Match the function with its graph.

1. $f\left(x\right)=3$

To determine

To find the domainof the function $f\left(x\right)=3$ .

Answer to Problem 1PRE

The domain of function $f\left(x\right)=3$ is $-\infty <x<\infty$ .

Explanation of Solution

Given: The function $f\left(x\right)=3$

Formula Used:

The domain of a function is the set of input or argument values for which the function is real and defined.

Calculation:

The function has no undefined points nor domain constraints. So, the domain will be all real numbers.

Therefore, the domain is $-\infty <x<\infty$ .

Conclusion:

The domain of function $f\left(x\right)=3$ is $-\infty <x<\infty$ .

To determine

To find the range of the function $f\left(x\right)=3$ :

Answer to Problem 1PRE

The range of function $f\left(x\right)=3$ is $\left\{3\right\}$ .

Explanation of Solution

Given: The function $f\left(x\right)=3$ .

Formula Used:

The range of a function is the set of values of the dependent variable for which a function is defined.

Calculation:

Here, the function $f\left(x\right)=3$ is a constant function. That means, for all input value of x, the output will be 3. So, the range of the given function is $\left\{3\right\}$ .

Conclusion:

The range of function $f\left(x\right)=3$ is $\left\{3\right\}$ .

To determine

To find x-intercept of the function $f\left(x\right)=3$ :

Answer to Problem 1PRE

There is no x-intercept

Explanation of Solution

Given: Function - $f\left(x\right)=3$

Formula Used:

x-intercept is a point on the graph where $y=0$

Calculation:

The y-intercept of a function is obtained at the point when $y=0$ . For the given function $f\left(x\right)=3$ ,

  $y=0\Rightarrow f\left(x\right)=0\Rightarrow 3=0$

But, $3\ne 0$ .

So, there is no such value of x which gives $f\left(x\right)=0$ . Hence, x-intercept does not exist.

Conclusion:

Hence, there are no x-axis interception points.

To determine

To find the y-intercept of the function $f\left(x\right)=3$ :

Answer to Problem 1PRE

The y-intercept of function $f\left(x\right)=3$ is $\left(0,3\right)$

Explanation of Solution

Given: Function - $f\left(x\right)=3$

Formula Used:

y-intercept is a point on the graph where $x=0$

Calculation:

Function is given as $f\left(x\right)=3$

For y-intercept, $x=0$ ;

And $f\left(0\right)=3$ ;

Thus, y-axis interception point is $\left(0,3\right)$

Conclusion:

The y-intercept of function $f\left(x\right)=3$ is $\left(0,3\right)$

To determine

To find Asymptotes (if applicable) of the function $f\left(x\right)=3$ :

Answer to Problem 1PRE

There are no asymptotes of the function $f\left(x\right)=3$

Explanation of Solution

Given: Function - $f\left(x\right)=3$

Formula Used:

If $f^{\prime }\left(x\right)>0$ , then f is increasing on the interval, and if $f^{\prime }\left(x\right)<0$ , then f is decreasing on the interval.

Calculation:

Given function is $f\left(x\right)=3$ .

There are no asymptotes as polynomial functions of degree 1 or higher can’t have asymptotes.

Conclusion:

Hence, there is no asymptote for the function

To determine

To find intervals over which the function $f\left(x\right)=3$ is increasing.

Answer to Problem 1PRE

Function is not an increasing function.

Explanation of Solution

Given: Function - $f\left(x\right)=3$

Formula Used:

If $f^{\prime }\left(x\right)>0$ , then f is increasing on the interval, and if $f^{\prime }\left(x\right)<0$ , then f is decreasing on the interval.

Calculation:

Derivative of $f\left(x\right)=3$ is

  $f^{\prime }\left(x\right)=0$ ; which is not an increasing function.

Thus, the function is not increasing.

Conclusion:

Hence, function is not an increasing function

To determine

To find intervals over which the function $f\left(x\right)=3$ is decreasing.

Answer to Problem 1PRE

Function is not a decreasing function.

Explanation of Solution

Given: Function - $f\left(x\right)=3$

Formula Used:

If $f^{\prime }\left(x\right)>0$ , then f is increasing on the interval, and if $f^{\prime }\left(x\right)<0$ , then f is decreasing on the interval.

Calculation:

Derivative of $f\left(x\right)=3$ is

  $f^{\text{'}}\left(x\right)=0$ ; which is not a decreasing function.

Thus, the function is not decreasing.

Conclusion:

Hence, the function is not a decreasing function

To determine

To graph the function $f\left(x\right)=3$ with its graph.

Explanation of Solution

Given: Function - $f\left(x\right)=3$

Graph:

Given function is $f\left(x\right)=3$

When $x=0$ , $f\left(0\right)=3$ ; then point $\left(0,3\right)$

When $y=0$ , $x$ is undefined

  

Thus, the graph is matched with the function