College Algebra: Functions, Graphs, and Their Properties

Functions and Relations

Definition of a Function

A function is a relation in which each element of the domain (input) is paired with exactly one element of the range (output). Not all relations are functions.

• Domain: The set of all possible input values (usually x-values).

• Range: The set of all possible output values (usually y-values).

Example: The relation {(-4, 19), (3, 10), (0, 1), (3, 11), (5, 23)} is not a function because the input 3 is paired with two different outputs (10 and 11).

Determining if an Equation Defines y as a Function of x

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

• Example: $4x - x^2 = y + 24$

• Solve for y: $y = 4x - x^2 - 24$

• For each x, there is only one y, so this is a function.

Evaluating and Combining Functions

Evaluating Functions

To evaluate a function, substitute the given value into the function's formula.

• Example: If $f(x) = |x| + 2$, find $f(-1)$ and $f(5)$.

• $f(-1) = |-1| + 2 = 1 + 2 = 3$

• $f(5) = |5| + 2 = 5 + 2 = 7$

Combining Functions

Functions can be combined using addition, subtraction, multiplication, and division.

• Example: If $f(x) = 9x - 4$ and $g(x) = 2x + 6$, then:

• $(f + g)(x) = f(x) + g(x) = (9x - 4) + (2x + 6) = 11x + 2$

• $(f imes g)(x) = f(x) imes g(x) = (9x - 4)(2x + 6)$

Properties of Functions

Even and Odd Functions

A function is even if $f(-x) = f(x)$ for all x in the domain. It is odd if $f(-x) = -f(x)$ for all x in the domain. If neither condition holds, the function is neither even nor odd.

• Example: $f(x) = |x|$ is even because $f(-x) = |-x| = |x|$.

• Example: $f(x) = x^3$ is odd because $f(-x) = (-x)^3 = -x^3 = -f(x)$.

Graphs of Functions

Interpreting Graphs

Graphs can be used to determine the domain, range, intercepts, and symmetry of a function.

• Domain: The set of all x-values for which the function is defined.

• Range: The set of all y-values the function attains.

• x-intercept: Where the graph crosses the x-axis (y = 0).

• y-intercept: Where the graph crosses the y-axis (x = 0).

• Symmetry: Even functions are symmetric about the y-axis; odd functions are symmetric about the origin.

Example: Analyzing a Graph

Given a graph (see Figure 1 in the original document), you may be asked to:

• Find the domain and range by observing the extent of the graph along the x- and y-axes.

• Identify intercepts by finding where the graph crosses the axes.

• Check for symmetry by folding the graph along the y-axis or rotating it 180° about the origin.

Graphing Absolute Value and Quadratic Functions

• Absolute Value Function: $f(x) = |x|$ produces a 'V' shaped graph with vertex at the origin.

• Quadratic Function: $f(x) = ax^2 + bx + c$ produces a parabola. The vertex, axis of symmetry, and intercepts can be found using algebraic methods.

Rates of Change and Slope

Average Rate of Change

The average rate of change of a function $f(x)$ from $x = a$ to $x = b$ is given by:

$\frac{f(b) - f(a)}{b - a}$

• This measures how much the function changes per unit increase in x.

• Example: For $f(x) = 4x + 11$, the average rate of change from $x = 2$ to $x = 4$ is:

• $\frac{f(4) - f(2)}{4 - 2} = \frac{(4 \times 4 + 11) - (4 \times 2 + 11)}{2} = \frac{27 - 19}{2} = 4$

Slope of a Line

The slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

• Example: For points (2, 4) and (1, 9): $m = \frac{9 - 4}{1 - 2} = \frac{5}{-1} = -5$

Using Graphs to Analyze Functions

Increasing, Decreasing, and Constant Intervals

A function is:

• Increasing on intervals where as x increases, f(x) increases.

• Decreasing on intervals where as x increases, f(x) decreases.

• Constant on intervals where as x increases, f(x) remains the same.

Example: Using Figure 2 (a parabola), the function increases on $(-\infty, 0.5)$ and decreases on $(0.5, \infty)$.

Finding Specific Function Values from a Graph

• To find x such that $f(x) = 8.25$, locate the y-value 8.25 on the graph and read the corresponding x-values.

• Example: In Figure 2, $f(0.5) = 8.25$.

Table: Properties of Even and Odd Functions

Property	Even Function	Odd Function
Algebraic Test	$f(-x) = f(x)$	$f(-x) = -f(x)$
Graph Symmetry	y-axis	Origin
Example	$f(x) = x^2$	$f(x) = x^3$

Graphing Piecewise and Absolute Value Functions

• Piecewise functions are defined by different expressions over different intervals.

• Absolute value functions can be written as $f(x) = |x|$, $f(x) = |x| - 7$, etc.

• To graph $f(x) = |x| - 7$, shift the basic absolute value graph down by 7 units.

Summary

• Understand the definition and properties of functions.

• Be able to evaluate, combine, and classify functions as even, odd, or neither.

• Use graphs to determine domain, range, intercepts, and symmetry.

• Calculate average rate of change and slope between points.

• Interpret and graph absolute value and quadratic functions.