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Lab Manual for MTH 111: Precalculus I: Functions

Mathematics Faculty

Section 1.2 Domain, Range, and Behaviors of Functions

In this section, we’ll learn to identify the domain and range of functions given in various forms, as well as determine when a function exhibits important behaviors.

Subsection Textbook Reference

This relates to content in §3.2 and §3.3 of Algebra and Trigonometry 2e
 1 
http://tiny.cc/111Z-Textbook
.

Exercises Preparation Exercises

1.

(a)
For any real number \(k\) other than \(0\text{,}\) what is \(\dfrac{0}{k}\) and why?
(b)
For any real number \(k\) other than \(0\text{,}\) what is \(\dfrac{k}{0}\) and why?
(c)
Given \(f(x) = \sqrt{x+2}\text{,}\) evaluate \(f(23)\) and \(f(-18)\text{.}\)

2.

(a)
Draw \(x \gt -2\) on a number line and write the set of values in both interval and set-builder notations.
Hint.
Here is a review video
 2 
https://youtu.be/aLvRu8Int4M
of these two notations.
(b)
Draw \(x \leq 6\) on a number line and write the set of values in both interval and set-builder notations.

Exercises Practice Exercises

1.

Algebraically find the domain of the following functions. State the domains in both interval notation and set-builder notation.
(a)
\(f(x) = \sqrt{-2x + 18}\)
(b)
\(g(t) = \sqrt[3]{3t-24}\)
(c)
\(h(k) = \dfrac{k+3}{k-9}\)

2.

Find the domain and range of the function \(p\) graphed in Figure 1.2.1. State both in interval notation and set-builder notation.
described in detail following the image
A curve beginning with an empty circle at \((-6,-5)\text{,}\) traveling up to just below \(y=-3\text{,}\) down to just below \(y=-4\text{,}\) then up to the point \((5,6)\text{.}\) There is an empty circle at \((-6,-5)\) and a filled circle at \((5,6)\text{.}\)
Figure 1.2.1. \(y=p(x)\)

3.

Below are the graphs of \(y=r(t)\) in Figure 1.2.2 and \(y=s(t)\) in Figure 1.2.3.
described in detail following the image
Graph of a function. The curve begins with a filled circle at \((-7,7)\text{,}\) travels smoothly to the point \((-4,1)\text{,}\) remains flat to the point \((-3,1)\text{,}\) then travels smoothly through the points \((-3,1)\text{,}\) \((-1,3)\text{,}\) \((3,-3)\text{,}\) before ending with an empty circle at the point \((6,2)\text{.}\)
Figure 1.2.2. \(y=r(t)\)
described in detail following the image
Graph of a continuous, irregular curve. It travels through the points \((-7,-6)\text{;}\) \((-6,-5)\text{;}\) \((-4,-1)\text{;}\) \((-3,0)\text{;}\) \((-1,-1)\text{;}\) \((0,-3)\text{;}\) \((1,-4)\text{;}\) \((2,-3)\text{;}\) \((3,-1)\text{;}\) \((4,0)\text{;}\) \((5,3)\text{;}\) \((6,3)\text{.}\) At the point \((-7,-6)\) there is an empty circle. At the point \((6,3)\) there is a filled circle.
Figure 1.2.3. \(y=s(t))\)
(a)
Over what intervals is \(r\) increasing?
(b)
Over what interval is \(s\) negative?
(c)
What is the absolute maximum value of \(r\text{?}\)
(d)
State any local minimum points of \(s\text{.}\)
(e)
Over what intervals is \(r\) constant?
(f)
Over what intervals is \(s\) decreasing?
(g)
Over what interval is \(r\) positive?
(h)
What is the absolute minimum value of \(s\text{?}\)

Subsection Definitions

Definition 1.2.4. Domain and Range.

The domain of a function is the set of all possible input values for the function.
The range of a function is the set of all possible output values for the function.
The domain and range are commonly stated using interval notation or set-builder notation.

Example 1.2.5. Domain and Range.

View this Desmos graph
 3 
https://tiny.cc/111Z-DomRang
to see an interactive example of these definitions.

Definition 1.2.6. Positive and Negative.

A function \(f\) is positive if the output values are greater than \(0\text{.}\) \(f\) is positive when \(f(x) \gt 0\text{.}\)
A function \(f\) is negative if the output values are less than \(0\text{.}\) \(f\) is negative when \(f(x) \lt 0\text{.}\)

Example 1.2.7. Positive and Negative.

View this Desmos graph
 4 
https://tiny.cc/111Z-PosNeg
to see an interactive example of these definitions.

Definition 1.2.8. Increasing, Decreasing, and Constant.

Let \(f\) be a function that is defined on an open interval \(I\text{,}\) with \(a\) and \(b\) in \(I\) and \(b \gt a\text{.}\)
\(f\) is increasing on \(I\) if \(f(b) \gt f(a)\) for all \(a\) and \(b\) in \(I\text{.}\) In other words, as you move left-to-right on the interval \(I\text{,}\) your \(y\)-values increase.
\(f\) is decreasing on \(I\) if \(f(b) \lt f(a)\) for all \(a\) and \(b\) in \(I\text{.}\) In other words, as you move left-to-right on the interval \(I\text{,}\) your \(y\)-values decrease.
\(f\) is constant on \(I\) if \(f(b) = f(a)\) for all \(a\) and \(b\) in \(I\text{.}\) In other words, as you move left-to-right on the interval \(I\text{,}\) your \(y\)-values do not change.

Example 1.2.9. Positive and Negative.

View this Desmos graph
 5 
https://tiny.cc/111Z-IncDec
to see an interactive example of these definitions.

Definition 1.2.10. Local Minimum or Maximum.

Given a function \(f\) that is defined on an open interval \(I\text{,}\) with \(c\) in \(I\text{.}\)
\(f\) has a local maximum at \(x = c\) if \(f(c) \geq f(x)\) for all \(x\) in \(I\text{.}\) The local maximum value of \(f\) is the output \(f(c)\text{.}\)
\(f\) has a local minimum at \(x = c\) if \(f(c) \leq f(x)\) for all \(x\) in \(I\text{.}\) The local minimum value of \(f\) is the output \(f(c)\text{.}\)

Example 1.2.11.

described in detail following the image
Graph of a smooth curve. The function begins near around \((-4, 4.8)\text{,}\) curves downward to a local minimum at \(x = -3\text{,}\) upward to a local maximum at around \(x=-1\text{,}\) downward to a local minimum at \(x = 2\text{,}\) then back up to around the point \((3, 1)\text{.}\) Each local minimum or maximum is at a peak or valley of the curve, and the local minimum at \(x = 2\) is much lower than the local minimum at \(x = -3\)
Figure 1.2.12. Local Extrema
In Figure 1.2.12, the function has two local minimum points and one local maximum point.
  • The local minimum value of about \(0.9\) occurs at \(x = -3\text{.}\)
  • The local minimum value of about \(-4.3\) occurs at \(x = 2\text{.}\)
  • The local maximum value of about \(3.5\) occurs at \(x = -1\text{.}\)

Definition 1.2.13. Absolute Minimum or Maximum.

\(f\) has an absolute maximum at \(x = c\) if \(f(c) \geq f(x)\) for all \(x\) in the domain of \(f\text{.}\) The absolute maximum value of \(f\) is the output \(f(c)\text{.}\)
\(f\) has an absolute minimum at \(x = c\) if \(f(c) \leq f(x)\) for all \(x\) in the domain of \(f\text{.}\) The absolute minimum value of \(f\) is the output \(f(c)\text{.}\)

Example 1.2.14.

described in detail following the image
Graph of the same curve from Figure 1.2.12. The point at \((-4,4.8)\) is the absolute maximum, sitting above all other points on the graph. The point at \((2,-4.3)\) is not only a local minimum, but also an absolute minimum, sitting below all other points on the graph.
Figure 1.2.15. Absolute Extrema
In Figure 1.2.15, the function has an absolute minimum point and an absolute maximum point.
  • The absolute minimum value of about \(-4.3\) occurs at \(x = 2\text{.}\)
  • The absolute maximum value of about \(4.8\) occurs at \(x = -4\text{.}\)

Exercises Exit Exercises

1.

What is the domain of \(f(x) = \sqrt{x}\text{?}\) What is the domain of \(g(x) = \sqrt[3]{x}\text{?}\) Why are these domains different?

2.

Graphically speaking, what is the difference between a function being negative and a function decreasing?

3.

For the function \(F\) graphed in Figure 1.2.16, answer the following.
described in detail following the image
Graph of a function. From below, the curve travels up to the point \((-7,2)\text{,}\) then passes smoothly through the points \((-4,-1)\text{;}\) \((-2,-1)\text{;}\) \((0,-3)\text{;}\) \((5,4)\text{;}\) \((7,0)\text{;}\) then continues downward and to the right off of the grid.
Figure 1.2.16. \(y=F(x)\)
(a)
Over what intervals is \(F\) increasing?
(b)
What is the range of \(F\text{?}\)
(c)
Over what intervals is \(F\) negative?
(d)
What are any local minimum points on \(F\text{?}\)
(e)
Over what intervals is \(F\) constant?
(f)
What is the absolute maximum value of \(F\text{?}\)

Reflection Reflection

1.

On a scale of 1-5, how are you feeling with the concepts related to the graphical behaviors of functions?
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