Algebra Study Guide: Linear Equations, Functions, Compositions, and Inverses

Vocabulary flashcards reviewing core definitions, formulas, and concepts from Chapters 3.3–3.5, 4.1, 5.3, and 5.4.

x-intercept

The point where a graph crosses the x-axis, found by setting $y = 0$ and written in the coordinate form $(x, 0)$.

y-intercept

The point where a graph crosses the y-axis, found by setting $x = 0$ and written in the coordinate form $(0, y)$.

Slope ($m$)

A measure of the steepness of a line, defined as $\text{Slope} = \frac{\text{rise}}{\text{run}}$ or calculated between two points using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.

Slope-Intercept Form

The equation of a line written as $y = mx + b$, where $m$ represents the slope and $b$ represents the y-intercept.

Point-Slope Form

The linear equation formula $y - y_1 = m(x - x_1)$, used when the slope $m$ and one point $(x_1, y_1)$ are known.

Parallel Lines

Lines that never intersect because they share the exact same slope ($m_1 = m_2$).

Perpendicular Lines

Lines that meet at a $90^\text{o}$ angle and have slopes that are negative reciprocals of each other, satisfying $m_1 m_2 = -1$.

Relation

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

Domain

The set of all first components or $x$-values (inputs) of a relation or function.

Range

The set of all second components or $y$-values (outputs) of a relation or function.

Function

A relation in which each input value has exactly one output value.

Vertical Line Test

A visual method used on a graph to verify if it represents a function: if any vertical line intersects the graph more than once, it is not a function.

Fraction Domain Restriction

A restriction stating that values making the denominator of a fraction equal to zero must be excluded from the domain ($\text{denominator} \neq 0$).

Square-Root Domain Restriction

A restriction stating that the expression located inside a square root must be greater than or equal to zero ($\text{inside} \geq 0$).

Difference Quotient

The formula used to compute the average rate of change of a function, given by $\frac{f(x + h) - f(x)}{h}$.

Function Composition

The operation of placing one function inside another, written as $(f \circ g)(x) = f(g(x))$ and evaluated from right to left (inside first).

Inverse Function ($f^{-1}(x)$)

A function that undoes the operation of another function, found algebraically by switching the $x$ and $y$ variables and solving for $y$.

One-to-One Function

A function where each output corresponds to only one input, which is required for the function's inverse to also be a function.

Horizontal Line Test

A graphical method used to check if a function is one-to-one: if every horizontal line intersects the graph at most once, the function is one-to-one.