Functions and Their Representations: College Algebra Study Notes

Functions and Their Representations

Definition of a Function

A function is a relation in which each input has exactly one output. In other words, for every element in the domain, there is a unique element in the range.

• Input: The independent variable, often denoted as x.

• Output: The dependent variable, often denoted as y or f(x).

Example: The table below shows the median income for different states. Each state (input) is assigned exactly one median income (output), so this is a function.

State	Median Income
Alabama	55
Indiana	65
Alaska	72
Colorado	83
Illinois	83
Massachusetts	83

Domain and Range

The domain of a function is the set of all possible input values (x-values). The range is the set of all possible output values (y-values).

• Domain: All valid inputs for the function.

• Range: All possible outputs the function can produce.

Example: For the Median Income function above, the domain is the set of states listed, and the range is the set {55, 65, 72, 83}.

Functions Defined Numerically

Functions can be represented using tables of values. To determine if a table represents a function, check that each input value corresponds to only one output value.

x	1	3	5
y	2	6	9	7

• If any input value is paired with more than one output, the relation is not a function.

Functions Defined by Tables: Example

Speed of vehicle (MPH)	30	40	50	60	70
Total stopping distance (feet)	109	173	253	346	454

• Function notation: $f(x) = y$

• Example: $StopDistance(40) = 173$

• To find $x$ such that $StopDistance(x) = 454$, look for the input value that gives 454 as output (here, $x = 70$).

Functions Defined by Graphs

Functions can also be represented graphically. The graph of a function passes the vertical line test: any vertical line should intersect the graph at most once.

Example: The graph below shows the stopping distance as a function of speed:

• To estimate $StopDistance(55)$, find the y-value on the graph where $x = 55$.

Interval Notation and Inequalities

Interval notation is used to describe sets of numbers, especially domains and ranges.

• Symbols: <, > use parentheses ( ), ≤, ≥ use brackets [ ]

• Examples:

Interval	Notation
$a \leq x \leq b$	$[a, b]$
$a < x < b$	$(a, b)$
$a < x \leq b$	$(a, b]$
$a \leq x < b$	$[a, b)$

• Example: $-4 < x \leq 8$ is written as $(-4, 8]$ in interval notation.

Evaluating Functions from Graphs

Given a graph, you can find the value of a function at a specific input by locating the corresponding point on the graph.

• Example: If $f(-1) = 2$, then the point $(-1, 2)$ is on the graph.

• The domain is the set of all x-values for which the graph exists.

• The range is the set of all y-values the graph attains.

Graphing by Hand

To graph a function by hand:

1. Make a table of values for several x-values.

2. Plot the corresponding points $(x, f(x))$.

3. Connect the points smoothly if the function is continuous.

• Example: For $f(x) = |x + 1|$, choose values for $x$, compute $f(x)$, and plot the points.

The Vertical Line Test

The vertical line test is a graphical method to determine if a curve is the graph of a function. If any vertical line crosses the graph more than once, the graph does not represent a function.

• Example: A parabola opening up or down passes the test; a circle does not.

Verbal and Symbolic Functions

Functions can be described in words (verbal), symbols (algebraic), or tables/graphs.

• Verbal Example: The function Complement takes an angle and subtracts it from 90°: $Complement(x) = 90 - x$.

• Symbolic Example: For $f(x) = -5x + 1$, to find $f(-3)$, substitute $x = -3$:

$f(-3) = -5(-3) + 1 = 15 + 1 = 16$

Determining if an Equation is a Function

To determine if an equation defines $y$ as a function of $x$, solve for $y$ and check if each $x$ gives only one $y$.

• Example: $y = 2x - 3$ is a function of $x$.

• $y^2 = x$ is not a function of $x$ because each $x$ gives two possible $y$ values.

Application Problems

Functions are used to model real-world situations, such as distance, speed, and growth.

• Example: The distance between two bicyclists as a function of time can be graphed and analyzed to answer questions about their motion.

• To evaluate a function at a specific value, substitute the value into the function's formula.

Summary Table: Types of Function Representations

Representation	Description	Example
Verbal	Describes the function in words	"Subtract the input from 90"
Numerical	Table of input-output pairs	See stopping distance table above
Graphical	Graph of the function	Stopping distance vs. speed graph
Symbolic	Algebraic formula	$f(x) = 2x - 3$

Additional info: The notes above include expanded explanations, examples, and context to ensure a self-contained study guide suitable for College Algebra students.