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Workbook for Stewart’s "Calculus"

Cory Wilson

Section 1.1 Four Ways to Represent a Function

Subsection 1.1.1 Before Class

https://mymedia.ou.edu/media/1.1-1/1_hbkr83p7
Figure 1. Pre-Class Video 1
https://mymedia.ou.edu/media/1.1-2/1_88q73nod
Figure 2. Pre-Class Video 2

Subsubsection 1.1.1.1 Functions

A relation is a rule which links an input variable to an output; given one piece of information, we can determine the corresponding piece. A special type of relation is one called a function.
Definition 1.1.1. Function/Domain/Range.
A function \(f\) is a rule that assigns to each element \(x\) in a set \(D\) one element, called \(f(x)\text{,}\) in a set \(E\text{.}\) The set \(D\) is called the domain of the function. The range of \(f\) is the set of all possible values of \(f(x)\) as \(x\) varies throughout the domain.
Definition 1.1.2. Independent Variable/Dependent Variable.
A symbol that represents an arbitrary element in the domain is called an independent variable. A symbol representing an arbitarary element in the range of a function is called a dependent variable.
Example 1.1.3.
Let \(C(t)\) represent the number of courses offered campus-wide during the week at time \(t\text{,}\) and \(O(t)\) represent the number of students walking on the South Oval at time \(t\) last Monday. Is \(C\) a function? What about \(O\text{?}\)
Solution.
\(C\) is not a function because at least one input has multiple outputs (for example, Monday at 12:00pm has multiple classes). \(O\) is a function because at time \(t\) last Monday specifically, there is a only one number of students on the South Oval.
Example 1.1.4.
Fill out the table with the domain and range of the given function. Write your answer in interval notation.
Function Domain Range
\(\sqrt{2+x}\)
\(\dfrac{x^2-1}{x-1}\)
\(x^3-6.2x^2+x-1\)
Solution.
Function Domain Range
\(\sqrt{2+x}\) \([-2,\infty)\) \([0,\infty]\)
\(\dfrac{x^2-1}{x-1}\) \((-\infty,1)\cup (1,\infty)\) \((-\infty,2)\cup (2,\infty)\)
\(x^3-6.2x^2+x-1\) \((-\infty,\infty)\) \((-\infty,\infty)\)
Example 1.1.5.
Assuming \(h\neq 0\text{,}\) simplify the difference quotient \(\dfrac{f(1+h)-f(1)}{h}\) where \(f(x) = 3x^2 -2x + 1\)
Solution.
The difference quotient simplifies to \(6+3h-2\)

Subsubsection 1.1.1.2 Representations of Functions

In mathematics, particularly applied mathematics, we need to be able to interpret real-world phenomena in four ways: numerically, algebraically, verbally, and graphically.
Example 1.1.6.
The price of gas at a certain 7-11 in Norman was $4.37 per gallon on June 26th.
  1. Is this information presented numerically, algebraically, verbally, or graphically?
  2. Represent this situation in the other three ways.
Solution.
  1. Verbally
  2. Numerically:
    Gallons 1 2 3 4 5
    Price 4.37 8.74 13.11 17.48 21.85
    Algebraically: \(p(g) = 4.37g\)
    Graphically:
    This is the graph of \(p(g) = 4.37g\) on the interval \([0,1]\)
Example 1.1.7.
A rectangular storage container has an open top, and a volume of 20 m\(^3\text{.}\) The length of its base is twice its width. Material for the base costs $5 per square meter; material for the sides costs $3 per square meter. Express the cost of materials as a function of the width of the base.
Solution.
\(C(w) = \dfrac{180}{w}+10w^2\) dollars
Vertical Line Test.
A curve in the \(xy-\)plane is the graph of a function of \(x\) if and only if no vertical line intersects the curve more than once.
Example 1.1.8.
Are both of these graphs functions? Why or why not?
The image on the left is the graph of the curve \(\cos x\) on the interval \([0,2\pi]\text{.}\)
The image on the right is the graph of the curve whose \(x-\)coordinate is given by the rule \(x^3-3x\text{,}\) and whose \(y-\)coordinate is given by the rule \(3x^2-9\text{.}\)
Solution.
The first graph is a function because it passes the vertical line test, but the second is not a function because it fails the vertical line test between \([-2,2]\text{.}\)
Example 1.1.9.
Below are numerical expressions for the relations \(h\) and \(k\text{.}\) Is \(h\) a function? What about \(k\text{?}\)
\(x\) 0 1 1 2 5 6
\(h(x)\) 0 1 2 3 4 5
\(t\) 0 1 1 2 5 6
\(k(t)\) 0 1 1 3 4 5
Solution.
\(k(t)\) is a function, but \(h\) is not.

Subsection 1.1.2 Pre-Class Activities

Example 1.1.10.

If \(f(x) = 6x^2 -3x + 1\text{,}\) find the following: \(f(1)\text{,}\) \(f(-2)\text{,}\) \(f(a)\text{,}\) \(f(-a)\text{,}\) \(f(a+1)\text{,}\) \(2f(a)\text{,}\) \(f(2a)\text{,}\) \(f(a^2)\text{,}\) \([f(a)]^2\text{,}\) and \(f(a+h)\text{.}\)
Solution.
\(f(1) = 4, f(-2) = 31, f(a) = 6a^2-3a+1, f(-a) = 6a^2+3a+1, f(a+1) = 6a^2+9a+4, 2f(a) = 12a^2-6a+2\)
\(f(2a) = 24a^2-6a+1, f(a^2)=6a^4-3a^2+1, [f(a)]^2 = 36a^4-36a^3+21a^2-6a+1, f(a+h) = 6a^2+12ah+6h^2-3a-3h+1\)

Example 1.1.11.

Evaluate the difference quotient \(\dfrac{f(x) - f(3)}{x-3}\) for \(f(x) = x^3\text{.}\)
Solution.
\(x^2+3x+9\)

Example 1.1.12.

Find the domain of the functions below:
  1. \(\displaystyle f(x) = \dfrac{3x^4 - 5}{x^2 +2x - 8}\)
  2. \(\displaystyle g(k) = \sqrt[3]{1-7k}\)
  3. \(\displaystyle h(t) = \sqrt{2-t} - \sqrt{3+t}\)
Solution.
  1. \(\displaystyle (-\infty,-4)\cup (-4,2)\cup (2,\infty)\)
  2. \(\displaystyle (-\infty,\infty)\)
  3. \(\displaystyle [-3,2]\)

Example 1.1.13.

Without referring to the vertical line test, explain why the graph below is not a function.
The image on the right is the graph of the curve whose \(x-\)coordinate is given by the rule \(x^2-3\text{,}\) and whose \(y-\)coordinate is given by the rule \(x^3-3x\text{.}\)
Solution.
Each input (except \(x=0\)) has multiple outputs

Subsection 1.1.3 In Class

Subsubsection 1.1.3.1 Piecewise Defined Functions

Definition 1.1.14.
A piecewise function is a function defined by different formulas in different parts of their domains.
Example 1.1.15.
A quick example of a piecewise function is the absolute value function:
\begin{equation*} f(x) = |x| = \begin{cases}-x \amp x \lt 0 \\x \amp x \geq 0 \end{cases} \end{equation*}
  1. What is \(f(-5)?\) What about \(f(1)\text{?}\)
  2. What is \(f(0)\text{?}\) Why?
  3. Sketch \(|x|\) on the interval \(-5\leq x \leq 5\text{.}\)
Solution.
  1. \(\displaystyle f(-5) = 5, f(1) = 1\)
  2. \(\displaystyle f(0) = 0\)
  3. Graph of \(f(x) = |x|\) on the interval \([-5,5]\)
Example 1.1.16.
A function \(h\) is defined by \(h(x) = \begin{cases}3-x \amp x \lt 2\\ x^2+x \amp x \geq 2 \end{cases}\)
  1. Evaluate \(h(-2)\text{,}\) \(h(3)\text{,}\) and \(h(2)\text{.}\)
  2. Sketch the graph of \(h\)
Solution.
  1. \(\displaystyle h(-2) = 5, h(3) = 12, h(2) = 6\)
  2. The graph of the piecewise function \(h(x)\) on the interval \([-2,4]\)
Example 1.1.17.
Write the absolute value function \(f(x) = |2x-3|\) as a piecewise function
Solution.
\(|2x-3| = \begin{cases} 3-2x \amp x \lt \dfrac{3}{2}\\ 2x-3 \amp x \geq \dfrac{3}{2} \end{cases}\)

Subsubsection 1.1.3.2 Symmetry

Definition 1.1.18. Even/Odd Function.
A function \(f\) is said to be even if it has the property that \(f(-x) = f(x)\text{.}\) A function \(g\) is said to be odd if it has the property that \(g(-x)=-g(x)\)
Example 1.1.19.
Determine if the following functions are even, odd, or neither.
  1. \(\displaystyle f(x) = x^7 + x^5 - x\)
  2. \(\displaystyle g(x) = 3-x^2\)
  3. \(\displaystyle h(x) = x^2 - x^3\)
Solution.
  1. Odd
  2. Even
  3. Neither

Subsubsection 1.1.3.3 Increasing and Decreasing Functions

Definition 1.1.20. Increasing/Decreasing.
Let \(f\) be a function defined on some input interval. \(f\) is said to be
  • increasing if the output values increase on the interval
  • decreasing if the output values decrease on the interval
Example 1.1.21.
Identify the intervals for which the function is increasing and decreasing.
This is the graph of the function \(\dfrac{1}{3}x^3 - \dfrac{1}{2}x^2 - 2x\) on the interval \([-3,4]\text{.}\)
Solution.
The function is increasing on \((-\infty,-1)\cup (2,\infty)\) and decreasing on \((-1,2)\)

Subsection 1.1.4 After Class Activities

Example 1.1.22.

Evaluate \(h(3), h(0),\) and \(h(2)\) for the function \(h(x) = \begin{cases}3 - \dfrac{1}{2}x \amp x \lt 2\\ 2x - 5 \amp x\geq 2 \end{cases}\text{.}\) Then, sketch the graph of \(h(x)\text{.}\)
Solution.
\(h(3) = 1, h(0) = 3, h(2) = -1\)
The graph of \(h(x)\) as described above

Example 1.1.23.

Sketch the graph of \(f(x) = x + |x|\) and \(g(x) = \begin{cases} |x| \amp |x| \leq 1\\ 1 \amp x \gt 1\end{cases}\text{.}\)
Solution.
The graph of \(f(x)\) as given above
The graph of \(g(x)\) as given above

Example 1.1.24.

If the point \((3,5)\) is on the graph of an even function, what other point must also be on the graph? What about if the function is odd? Justify your answers.
Solution.
The point \((-3,5)\) is on the even function, and the point \((-3,-5)\) is on the odd function.

Example 1.1.25.

Is the function \(f(t) = t|t|\) even, odd, or neither? Explain.
Solution.
The function is odd.

Subsection 1.1.5 Section 1.1 Additional Resources

Subsubsection 1.1.5.1 Functions

Subsubsection 1.1.5.2 Piecewise Functions

Subsubsection 1.1.5.3 Symmetry

Subsubsection 1.1.5.4 Difference Quotients

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