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Section 1.1 Functions and Their Graphs

This section of the book is a review of functions.

Subsection 1.1.1 Function Basics

Definition 1.1.1. Function.

A function \(f\) is a rule that assigns to every member \(x\) of a domain \(D\) a unique element \(f(x)\text{.}\)

Functions can be thought of as a “machine” or “black box” and can be represented/described in a variety of ways:

  • Verbal

  • Formula

  • Table

  • Graph

Example 1.1.2.

Express the edge length of a cube as a function of the cube's diagonal length d. Then express the surface area and volume of the cube as a function of the diagonal length, and graph the three functions.

Answer.

edge length: \(\displaystyle e(d) = \frac{d}{\sqrt{3}}\)

surface area: \(\displaystyle S(d) = 2d^2\)

volume: \(\displaystyle V(d) = \frac{d^3}{3\sqrt{3}}\)

A plot of the functions \(\displaystyle e(d) = \frac{d}{\sqrt{3}}\text{,}\) \(\displaystyle S(d) = 2d^2\text{,}\) and \(\displaystyle V(d) = \frac{d^3}{3\sqrt{3}}\text{.}\)
Figure 1.1.3. Edge length, surface area, and volume of a cube as functions of diagonal length.

Pay attention to notation: \(y=f(x)\)

  • \(x\) and \(y\) are variables representing real numbers (elements of the domain and codomain).

  • \(f\) is a variable representing a function.

  • \(f(x)\) is the result of applying the function \(f\) to the number represented by \(x\text{.}\)

  • Writing \(f(x)=2x^3-5\) expresses the result of applying the function \(f\) to the number represented by \(x\text{.}\)

  • While writing \(y=2x^3-5\) expresses a relationship between the variables \(x\) and \(y\text{.}\)

Definition 1.1.4.

The domain of a function is the set of all possible input values. For a function defined by a formula, the (natural) domain is the largest set of real numbers for which the formula gives a real number result.

Example 1.1.5.

  1. What is the (natural) domain of \(f(x)=130 - 5x\text{?}\)

  2. A tank initially holding 130 cubic feet of water is emptied by a pump that pumps 5 cubic feet of water a minute. The volume of water in the take after \(t\) minutes of pumping is \(V(t)=130 - 5t\text{.}\) What is the domain of V?

Answer.

\((-\infty,\infty)\)

\([0,26]\)

Example 1.1.6.

Write the following in interval notation.

  1. \(\displaystyle x\geq 5\)

  2. \(\displaystyle x \lt 3\)

  3. \(\displaystyle -3 \leq x \lt 7\)

  4. All real numbers neither between nor including \(-1\) and \(1\)

  5. All real numbers other than \(4\)

Answer.

  1. \(\displaystyle [5,\infty)\)

  2. \(\displaystyle (-\infty,3)\)

  3. \(\displaystyle [-3,7)\)

  4. \(\displaystyle (-\infty,-1) \cup (1,\infty)\)

  5. \(\displaystyle (-\infty,4) \cup (4,\infty)\)

Definition 1.1.7.

A function's codomain is a set containing all possible output values. The range of a function is the set of all output values actually obtained by the function.

Example 1.1.8.

What is the range of \(S(d)=2d^2\text{?}\)

Answer.

\([0,\infty)\)

For functions arising out of a specific context, the natural domain is not always realistic and a more restrictive domain may be more appropriate.

Example 1.1.9.

  1. What is the range of \(f(x)=130 - 5x\) (on its natural domain)?

  2. What is the range of \(V(t)=130 - 5t\text{,}\) restricted to the domain \([0,26]\text{?}\)

Subsection 1.1.2 Common Functions

See also the handout on common functions 1 .

Constant Functions.

A constant function is a function defined by \(f(x)=c\text{,}\) where \(c\) is any real number (possibly 0).

Formula

\(\displaystyle f(x)=c\)

Domain

\(\displaystyle (-\infty,\infty)\)

Range

\(\displaystyle \{c\}\)

Graph

A horizontal line.

A plot of the function \(f(x)=3\text{.}\)
Figure 1.1.11. \(f(x)=3\)

Linear Functions.

A linear function is a function of the form \(f(x)=mx+b\) where \(m\) and \(b\) are real numbers with \(m\neq 0\text{.}\)

Formula

\(f(x)=mx+b\text{,}\) with \(m\neq 0\)

Domain

\(\displaystyle (-\infty,\infty)\)

Range

\(\displaystyle (-\infty,\infty)\)

Graph

A line.

A plot of the function \(f(x)=2x+3\text{.}\)
Figure 1.1.12. \(f(x)=2x+3\)

The identity function is the linear function with \(m=1\) and \(b=0\text{,}\) i.e., \(f(x)=x\text{.}\)

Power Functions.

A power function is a function of the form \(f(x)=x^a\) for a constant \(a\text{.}\) The properties of the function varies significantly depending on the value of \(a\text{.}\)

\(a=1\).

The power function with exponent 1 is the identity function.

\(a\) a positive even integer.

When \(a=2,4,6,\ldots\text{,}\) the graph of a power function is U-shaped.

Formula

\(f(x)=x^a\text{,}\) with \(a=2,4,6,\ldots\)

Domain

\(\displaystyle (-\infty,\infty)\)

Range

\(\displaystyle [0,\infty)\)

Graph

U-shaped

A plot of the functions \(f(x)=x^2\text{,}\) \(f(x)=x^4\text{,}\) \(f(x)=x^6\text{.}\)
Figure 1.1.13. \(f(x)=x^2\text{,}\) \(f(x)=x^4\text{,}\) \(f(x)=x^6\)
\(a\) a positive odd integer.

When \(a=1,3,5,\ldots\text{,}\) the graph of a power function is a bent line.

Formula

\(f(x)=x^a\text{,}\) with \(a=1,3,5,\ldots\)

Domain

\(\displaystyle (-\infty,\infty)\)

Range

\(\displaystyle (-\infty,\infty)\)

Graph

bent line

A plot of the functions \(f(x)=x^1\text{,}\) \(f(x)=x^3\text{,}\) \(f(x)=x^5\text{.}\)
Figure 1.1.14. \(f(x)=x^1\text{,}\) \(f(x)=x^3\text{,}\) \(f(x)=x^5\)
\(a\) a negative even integer.

When \(a=-2,-4,-6,\ldots\text{,}\) the graph of a power function is a volcano.

Formula

\(f(x)=x^a\text{,}\) with \(a=-2,-4,-6,\ldots\)

Domain

\(\displaystyle (-\infty,0)\cup(0,\infty)\)

Range

\(\displaystyle (0,\infty)\)

Graph

volcano

A plot of the functions \(f(x)=x^{-2}\text{,}\) \(f(x)=x^{-4}\text{,}\) \(f(x)=x^{-6}\text{.}\)
Figure 1.1.15. \(f(x)=\frac{1}{x^2}\text{,}\) \(f(x)=\frac{1}{x^4}\text{,}\) \(f(x)=\frac{1}{x^6}\)
\(a\) a negative odd integer.

When \(a=-1,-3,-5,\ldots\text{,}\) the graph of a power function is not a volcano.

Formula

\(f(x)=x^a\text{,}\) with \(a=-1,-3,-5,\ldots\)

Domain

\(\displaystyle (-\infty,0)\cup(0,\infty)\)

Range

\(\displaystyle (-\infty,0)\cup(0,\infty)\)

Graph

?

A plot of the functions \(f(x)=x^{-1}\text{,}\) \(f(x)=x^{-3}\text{,}\) \(f(x)=x^{-5}\text{.}\)
Figure 1.1.16. \(f(x)=\frac{1}{x}\text{,}\) \(f(x)=\frac{1}{x^3}\text{,}\) \(f(x)=\frac{1}{x^5}\)
Even root.

When \(a=\frac{1}{2},\frac{1}{4},\frac{1}{6},\ldots\text{.}\)

Formula

\(f(x)=x^a\text{,}\) with \(a=\frac{1}{2},\frac{1}{4},\frac{1}{6},\ldots\)

Domain

\(\displaystyle [0,\infty)\)

Range

\(\displaystyle [0,\infty)\)

Graph

?

A plot of the functions \(f(x)=x^{1/2}\text{,}\) \(f(x)=x^{1/4}\text{,}\) \(f(x)=x^{1/6}\text{.}\)
Figure 1.1.17. \(f(x)=\sqrt{x}\text{,}\) \(f(x)=\sqrt[4]{x}\text{,}\) \(f(x)=\sqrt[6]{x}\)
Odd root.

When \(a=\frac{1}{3},\frac{1}{5},\frac{1}{7},\ldots\text{.}\)

Formula

\(f(x)=x^a\text{,}\) with \(a=\frac{1}{3},\frac{1}{5},\frac{1}{7},\ldots\)

Domain

\(\displaystyle (-\infty,\infty)\)

Range

\(\displaystyle (-\infty,\infty)\)

Graph

?

A plot of the functions \(f(x)=x^{1/3}\text{,}\) \(f(x)=x^{1/5}\text{,}\) \(f(x)=x^{1/7}\text{.}\)
Figure 1.1.18. \(f(x)=\sqrt[3]{x}\text{,}\) \(f(x)=\sqrt[5]{x}\text{,}\) \(f(x)=\sqrt[7]{x}\)

Polynomial Functions.

A polynomial function is a function of the form \(f(x)=a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\text{,}\) with \(n\) a positive integer and \(a_n\neq 0\text{.}\) The degree of the polynomial is the positive integer \(n\text{.}\)

Formula

\(\displaystyle f(x)=a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\)

Domain

\(\displaystyle (-\infty,\infty)\)

Range

\(\displaystyle Varies\)

Graph

A plot of the polynomial \(p(x)=2x+3\text{.}\)
Figure 1.1.19. Linear polynomial: \(p(x)=2x+3\)
A plot of the polynomial \(p(x)=2x^2-3x-1\text{.}\)
Figure 1.1.20. Quadratic polynomial: \(p(x)=2x^2-3x-1\)
A plot of the polynomial \(p(x)=x^3+2x^2-5x-6\text{.}\)
(a) \(p(x)=x^3+2x^2-5x-6\)
A plot of the polynomial \(p(x)=-x^3+2x^2+x+2\text{.}\)
(b) \(p(x)=-x^3+2x^2+x+2\)
Figure 1.1.21. Two cubic polynomials

Rational Functions.

A rational function is the ratio of two polynomials; that is, it is of the form \(f(x) = \frac{p(x)}{q(x)}\) where \(p\) and \(q\) are both polynomials.

Formula

\(\displaystyle f(x) = \frac{p(x)}{q(x)}\)

Domain

\(\displaystyle q(x)\neq 0\)

Range

\(\displaystyle Varies\)

Graph

A plot of the polynomial \(f(x)=\frac{x+2}{x^2-1}\text{.}\)
(a) \(f(x)=\frac{x+2}{x^2-1}\)
A plot of the polynomial \(f(x)=\frac{2x^2-3x-2}{x^2+1}\text{.}\)
(b) \(f(x)=\frac{2x^2-3x-2}{x^2+1}\)
A plot of the polynomial \(f(x)=\frac{3x^2+7x-6}{x+2}\text{.}\)
(c) \(f(x)=\frac{3x^2+7x-6}{x+2}\)
Figure 1.1.22. Three rational functions.

Definition 1.1.23. Algebraic function.

A function is algebraic if its formula involves only addition, subtraction, multiplication, division, and radicals.

Sine and Cosine.

Formula

\(f(x) = \sin(x)\text{,}\) \(f(x)=\cos(x)\)

Domain

\(\displaystyle (-\infty,\infty)\)

Range

\(\displaystyle [-1,1]\)

Graph

A plot of sine.
(a) \(f(x)=\sin(x)\)
A plot of cosine.
(b) \(f(x)=\cos(x)\)
Figure 1.1.24. Sine and Cosine.

Exponential Functions.

An exponential function is a constant raised to a variable power; that is, it is of the form \(f(x) = a^x\) where \(a\) is a real number such that \(a\gt 0\) and \(a\neq 1\text{.}\)

Formula

\(f(x) = a^x\) for \(a\gt 0\) and \(a\neq 1\)

Domain

\(\displaystyle (-\infty,\infty)\)

Range

\(\displaystyle (0,\infty)\)

Graph

A plot of the exponential function with base 2.
(a) \(f(x)=2^x\)
A plot of the exponential function with base 1/2.
(b) \(f(x)=\left(\frac{1}{2}\right)^x\)
Figure 1.1.25. Exponential functions.

Logarithmic Functions.

A logarithmic function is the inverse of the exponential. It is of the form \(f(x) = \log_a(x)\) where \(a\) is a real number such that \(a\gt 0\) and \(a\neq 1\text{.}\)

Formula

\(f(x) = \log_a(x)\) for \(a\gt 0\) and \(a\neq 1\)

Domain

\(\displaystyle (0,\infty)\)

Range

\(\displaystyle (-\infty,\infty)\)

Graph

A plot of the logarithmic function with base 2.
(a) \(f(x)=\log_2(x)\)
A plot of the logarithmic function with base 1/2.
(b) \(f(x)=\log_{1/2}(x)\)
Figure 1.1.26. Logarithmic functions.

Definition 1.1.27. Transcendental function.

A function is transcendental if it is not algebraic. Transcendental functions include the trigonometric, exponential, and logarithmic functions, among others.

Piecewise-Defined Functions.

A piecewise-defined function is a function that is defined by different formulas over separate intervals.

Example 1.1.28.

For the function

\begin{equation*} f(x)=\begin{cases} 2 \amp x\leq -5\\ x \amp -5 \lt x \lt 3\\ x^2-6 \amp x\geq 3 \end{cases} \end{equation*}

calculate \(f(-7), f(-5), f(-1), f(3), f(5)\) and sketch a graph.

Absolute Value Function.

The absolute value function is the function that takes every real number and returns its absolute value. It can be defined as

\begin{equation*} |x|=\begin{cases} x \amp x\geq 0\\ -x \amp x \lt 0 \end{cases} \end{equation*}
Formula

\(\displaystyle f(x)=|x|\)

Domain

\(\displaystyle (-\infty,\infty)\)

Range

\(\displaystyle [0,\infty)\)

Graph

V

A plot of the absolute value function.
Figure 1.1.29. \(f(x)=|x|\)

Subsection 1.1.3

Definition 1.1.30. Increasing/Decreasing.

A function \(f\) is said to be increasing on an interval \(I\) if the graph rises from left to right throughout that interval; i.e., if \(f(x_2)\gt f(x_1)\) whenever \(x_2\gt x_1\) are in \(I\text{.}\)

A function \(f\) is said to be decreasing on an interval \(I\) if the graph falls from left to right throughout that interval; i.e., if \(f(x_2)\lt f(x_1)\) whenever \(x_2\gt x_1\) are in \(I\text{.}\)

Note 1.1.31.

The interval on which a function is increasing or decreasing should (for MyMathLab's sake) always be open, e.g., we would say that \(y=x^2\) increases on \((0,\infty)\text{.}\)

Example 1.1.32.

Over what intervals are the standard functions above increasing and decreasing?

Definition 1.1.33. Even/Odd Function.

A function \(f\) is even if \(f(-x)=f(x)\text{;}\) that is, if it is symmetric about the \(y\)-axis.

A function \(f\) is odd if \(f(-x)=-f(x)\text{;}\) that is, if it is symmetric about the origin.

Example 1.1.34.

Which of the standard functions described earlier are even, which are odd, and which are neither?

Answer.

even functions include: constant functions, power functions of even (positive or negative) degree, \(\cos\text{,}\) absolute value

odd functions include: the identity function, power functions of odd (positive or negative) degree, \(n\)th root for odd \(n\text{,}\) \(\sin\)

Subsection 1.1.4 More on domain

Example 1.1.36.

Find the domains and ranges of:

  1. \(\displaystyle f(x)=\frac{5}{x+2}\)

  2. \(\displaystyle g(x)=\sqrt{x-7}\)

  3. \(\displaystyle h(x)=\sqrt{9-x^2}\)

Answer.

  1. \(\displaystyle (-\infty,-2)\cup (-2,\infty)\)

  2. \(\displaystyle [7,\infty)\)

  3. \(\displaystyle [-3,3]\)

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