Algebra Review: Linear Equations, Functions, and Inverses

Vocabulary flashcards covering linear equations, slopes, relations, functions, domain restrictions, composition of functions, and inverse functions based on the lecture notes.

$x$-intercept

The point where a graph crosses the $x$-axis, found by setting $y = 0$ and solving for $x$.

$y$-intercept

The point where a graph crosses the $y$-axis, found by setting $x = 0$ and solving for $y$.

Slope Formula

$m = \frac{y_2 - y_1}{x_2 - x_1}$, which describes a line's steepness.

Slope-Intercept Form

The linear equation form $y = mx + b$, where $m$ represents the slope and $b$ represents the $y$-intercept.

Point-Slope Form

The linear equation form $y - y_1 = m(x - x_1)$, used when a line's slope $m$ and a point $(x_1, y_1)$ are known.

Parallel Lines Slope Rule

A rule stating that parallel nonvertical lines have equal slopes.

Perpendicular Lines Slope Rule

A rule stating that nonvertical, nonhorizontal perpendicular lines have slopes that are negative reciprocals of each other, satisfying the equation $m_1 \cdot m_2 = -1$.

Relation

A set of ordered pairs $(x, y)$.

Domain

The set of all $x$-values or first coordinates in a relation.

Range

The set of all $y$-values or second coordinates in a relation.

Function

A relation where each input $x$ has only one output $y$.

Vertical Line Test

A graphical test used to determine whether a graph represents a function.

Horizontal Line Test

A graphical test used to check whether a function is one-to-one and has an inverse function.

Rational Function Domain Restriction

The condition requiring that the denominator of a rational function cannot equal zero.

Square Root Domain Restriction

The condition requiring that the expression inside a square root must be greater than or equal to zero.

Interval Notation Symbols

Notation where parentheses exclude endpoints and brackets include endpoints.

Difference Quotient

The formula $\frac{f(x + h) - f(x)}{h}$.

Function Composition

The operation denoted by $(f \circ g)(x) = f(g(x))$, which involves evaluating the inside function $g$ first and then applying $f$.

Inverse Function

A function that undoes the original function.

Three Steps to Find an Inverse

1. Replace $f(x)$ with $y$. 2. Switch $x$ and $y$. 3. Solve the equation for $y$.

Inverse Relation for Ordered Pairs

Switching the coordinates of each point so that $(x, y)$ becomes $(y, x)$.