College Algebra: Functions, Graphs, and Equations Study Guide

Functions and Relations

Definition and Identification

A function is a relation in which each input (x-value) corresponds to exactly one output (y-value). A relation is any set of ordered pairs.

• Function: Each x-value is paired with only one y-value.

• Not a function: An x-value is paired with more than one y-value.

• Example: The set {(1, -4), (2, -11), (6, -5), (8, 3), (10, 30)} is a function because all x-values are unique.

Evaluating Functions

Substitution

To evaluate a function at a given value, substitute the value for x in the function's formula.

• Example: If $f(x) = x^2 - 2x - 7$, then $f(3) = 3^2 - 2(3) - 7 = 9 - 6 - 7 = -4$.

• Example: If $f(x) = 4x^2 + 3x - 7$, then $f(2) = 4(2)^2 + 3(2) - 7 = 16 + 6 - 7 = 15$.

Transformations of Functions

Shifting and Reflecting Graphs

Transformations change the position or shape of a graph. Common transformations include shifting (translation) and reflecting.

• Horizontal shift: $f(x - h)$ shifts the graph h units to the right; $f(x + h)$ shifts h units to the left.

• Vertical shift: $f(x) + k$ shifts the graph k units up; $f(x) - k$ shifts k units down.

• Reflection: $-f(x)$ reflects the graph across the x-axis.

• Example: The graph of $f(x) = ext{sqrt}(x - 4)$ can be obtained from $y = ext{sqrt}(x)$ by shifting 4 units to the right.

Graphing and Function Properties

Vertical Line Test

The vertical line test determines if a graph represents a function. If any vertical line crosses the graph more than once, it is not a function.

• Example: A parabola opening upwards passes the vertical line test and is a function.

One-to-One Functions

A function is one-to-one if each y-value is paired with only one x-value. This can be checked using the horizontal line test.

• Example: $f(x) = x^3$ is one-to-one; $f(x) = x^2$ is not one-to-one.

Inverse Functions

Definition and Verification

Two functions $f(x)$ and $g(x)$ are inverses if $f(g(x)) = x$ and $g(f(x)) = x$ for all x in the domain.

• Example: $f(x) = rac{3x}{x^2 - 2}$ and $g(x) = rac{2x + 3}{x}$ are not inverses because $f(g(x)) eq x$.

Function Operations

Sum, Difference, Product, and Quotient

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

• Sum: $(f + g)(x) = f(x) + g(x)$

• Difference: $(f - g)(x) = f(x) - g(x)$

• Product: $(f imes g)(x) = f(x) imes g(x)$

• Quotient: $(f / g)(x) = rac{f(x)}{g(x)}$, $g(x) eq 0$

• Example: If $f(x) = x + 5$, $g(x) = x - 4$, then $(f + g)(x) = (x + 5) + (x - 4) = 2x + 1$.

Symmetry of Graphs

Types of Symmetry

Graphs may be symmetric with respect to the x-axis, y-axis, or origin.

• Y-axis symmetry: $f(-x) = f(x)$ (even function)

• Origin symmetry: $f(-x) = -f(x)$ (odd function)

• X-axis symmetry: Not a function

• Example: $y = x^2$ is symmetric about the y-axis; $y = x^3$ is symmetric about the origin.

Even and Odd Functions

Definitions

• Even function: $f(-x) = f(x)$ for all x in the domain.

• Odd function: $f(-x) = -f(x)$ for all x in the domain.

• Neither: If neither condition is met.

• Example: $f(x) = x^2$ is even; $f(x) = x^3$ is odd; $f(x) = x^2 + x$ is neither.

Piecewise Functions

Definition and Evaluation

A piecewise function is defined by different expressions for different intervals of the domain.

• Example: $f(x) = \begin{cases} 6 & \text{if } x \leq -1 \\ 4x & \text{if } 0 < x < 3 \\ x & \text{if } x \geq 3 \end{cases}$

Linear Equations and Graphs

Slope-Intercept and Point-Slope Forms

The equation of a line can be written in several forms:

• Slope-intercept form: $y = mx + b$, where m is the slope and b is the y-intercept.

• Point-slope form: $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line.

• Example: The line through (3, 5) with slope $-3$ is $y - 5 = -3(x - 3)$.

Parallel and Perpendicular Lines

• Parallel lines: Have the same slope.

• Perpendicular lines: Slopes are negative reciprocals.

• Example: A line parallel to $2x + y = 4$ through (5, 2) has the same slope as the given line.

Distance Between Points

Distance Formula

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

• $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Example: The distance between (7, 7) and (-6, -7) is $\sqrt{(7 - (-6))^2 + (7 - (-7))^2} = \sqrt{13^2 + 14^2} = \sqrt{169 + 196} = \sqrt{365}$.

Equations and Graphs of Circles

Standard Form

The equation of a circle with center $(h, k)$ and radius $r$ is:

• $(x - h)^2 + (y - k)^2 = r^2$

• Example: Center (-8, 6), radius 3: $(x + 8)^2 + (y - 6)^2 = 9$

Summary Table: Function Properties

Property	Definition	Example
Function	Each x maps to one y	$f(x) = x^2$
One-to-one	Each y maps to one x	$f(x) = x^3$
Even	$f(-x) = f(x)$	$f(x) = x^2$
Odd	$f(-x) = -f(x)$	$f(x) = x^3$
Neither	Does not satisfy even or odd	$f(x) = x^2 + x$

Additional info:

• Some questions and answers are presented in short form; full academic context has been added for clarity.

• Graph images referenced in the original file are described in text for accessibility.