Function Families: End Behavior, Domain, and Range

Vocabulary practice flashcards covering parent functions, graph recognition, domain, range, and end behavior for standard function families.

Domain

inputs ($x$)

Range

outputs ($y$)

Interval Notation: Parentheses ( )

Notation indicating that the endpoint is not included; always placed on positive infinity ($\infty$) and negative infinity ($-\infty$).

Interval Notation: Square Brackets [ ]

Notation indicating that the endpoint is included in the set.

End Behavior

A description of what values $f(x)$ approaches ($-\infty$, $\infty$, $0$, or a number) as $x$ approaches negative infinity ($x \rightarrow -\infty$) or positive infinity ($x \rightarrow \infty$).

Linear Function

A function family with the parent equation $f(x) = x$, having domain $(-\infty, \infty)$, range $(-\infty, \infty)$, and end behavior approaching $-\infty$ as $x \rightarrow -\infty$ and $\infty$ as $x \rightarrow \infty$.

Quadratic Function

$f(x) = x^2$, graph is a parabola with domain $(-\infty, \infty)$, range $[0, \infty)$ ($y \ge 0$), and both ends approaching $\infty$.

Cubic Function

A function family with the equation $f(x) = x^3$, having domain $(-\infty, \infty)$, range $(-\infty, \infty)$, and end behavior approaching $-\infty$ as $x \rightarrow -\infty$ and $\infty$ as $x \rightarrow \infty$.

Absolute Value Function

A function family with the equation $f(x) = |x|$, creating a V-shaped graph with domain $(-\infty, \infty)$, range $[0, \infty)$ ($y \ge 0$), and both ends approaching $\infty$.

Square Root Function

A function family with the equation $f(x) = \sqrt{x}$, having domain $[0, \infty)$ ($x \ge 0$) and range $[0, \infty)$ ($y \ge 0$).

Reciprocal Function

A function family with the equation $f(x) = \frac{1}{x}$, having domain $(-\infty, 0) \cup (0, \infty)$ and range $(-\infty, 0) \cup (0, \infty)$, where outputs approach $0$ as $x \rightarrow -\infty$ and as $x \rightarrow \infty$.

Exponential Function (base 2)

A function family with the equation $f(x) = 2^x$, having domain $(-\infty, \infty)$ and range $(0, \infty)$ ($y > 0$), where $f(x) \rightarrow 0$ as $x \rightarrow -\infty$ and $f(x) \rightarrow \infty$ as $x \rightarrow \infty$.

Quartic Function

A function family with the equation $f(x) = x^4$, with domain $(-\infty, \infty)$ and range $[0, \infty)$; compared to a quadratic graph, it is flatter near the origin and rises steeper for $|x| > 1$.

Quintic Function

A function family with the equation $f(x) = x^5$, which shares the exact same end behavior as the cubic function ($f(x) \rightarrow -\infty$ as $x \rightarrow -\infty$ and $f(x) \rightarrow \infty$ as $x \rightarrow \infty$).

Left End Behavior of the Square Root Function

Does not exist,

Algebraic Domain Restriction of $f(x) = \frac{1}{x}$

The input $0$ is excluded from the domain of the reciprocal function because dividing by $0$ is undefined.

Range Comparison: $f(x) = 2^x$ vs. $g(x) = x^2$

Neither function outputs negative values, but $x^2$ can equal $0$ (range $[0, \infty)$), whereas $2^x$ is strictly greater than $0$ and never reaches $0$ (range $(0, \infty)$).

Function Family of Graph A

Cubic function, with parent equation $f(x) = x^3$.

Function Family of Graph B

Reciprocal function, with parent equation $f(x) = \frac{1}{x}$.

Function Family of Graph C

Absolute value function, with parent equation $f(x) = |x|$.

Function Family of Graph D

Exponential function (base 2), with parent equation $f(x) = 2^x$.

Function Family of Graph E

Square root function, with parent equation $f(x) = \sqrt{x}$.

Function Family of Graph F

Linear function, with parent equation $f(x) = x$.

Function Family of Graph G

Quartic function, with parent equation $f(x) = x^4$.

Function Family of Graph H

Quadratic function, with parent equation $f(x) = x^2$.