College Algebra: Parent Functions and Function Notation

Vocabulary flashcards covering parent functions, their equations, domains, and ranges, as well as core concepts of relations, functions, and one-to-one functions.

Relation

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

Function

A relation that maps each input in the domain to exactly one output in the range (informally, no $x$'s repeat).

Independent Variable

The variable representing inputs in a relation or function, denoted by $x$.

Dependent Variable

The variable representing outputs in a relation or function, denoted by $y$.

One-to-One Function

A function in which each input has exactly one output and each output has exactly one input (informally, no $x$'s repeat and no $y$'s repeat).

The Birthday Example

A mapping analogy illustrating function rules: multiple people can share the same birthday (different inputs may have the same output), but each person must have exactly one birthday and cannot have more than one (each input maps to exactly one output).

(Increasing) Line Parent Function

The parent function defined by $y = x$ (or $y = 1x + 0$), with domain $(-\infty, \infty)$ and range $(-\infty, \infty)$.

Parabola Parent Function

The parent function defined by $y = x^2$, having a vertex at $(0, 0)$, domain $(-\infty, \infty)$, and range $[0, \infty)$.

Cubic Function Parent Function

The parent function defined by $y = x^3$, passing through $(0, 0)$, with domain $(-\infty, \infty)$ and range $(-\infty, \infty)$.

Square Root Function Parent Function

The parent function defined by $y = \sqrt{x}$, lying in Quadrant 1 with starting point $(0, 0)$, domain $[0, \infty)$, and range $[0, \infty)$.

Hyperbola Parent Function

The parent function defined by $y = \frac{1}{x}$, which does not touch the axes and passes through points like $(2, \frac{1}{2})$ and $(-3, -\frac{1}{3})$, with domain $(-\infty, 0) \cup (0, \infty)$ and range $(-\infty, 0) \cup (0, \infty)$.

Inverse-Square Function Parent Function

The parent function defined by $y = \frac{1}{x^2}$, which does not touch the axes and passes through points like $(2, \frac{1}{4})$ and $(3, \frac{1}{9})$, with domain $(-\infty, 0) \cup (0, \infty)$ and range $(0, \infty)$.

Unit Circle

A geometric relation centered at the origin defined by $x^2 + y^2 = 1$ with radius $r = 1$, domain $[-1, 1]$, and range $[-1, 1]$; it is not a function.

Equation of Any Circle

The standard form equation $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ represents the center of the circle and $r$ represents the radius.

Semicircle (centered at origin) Parent Function

The parent function defined by $y = \sqrt{r^2 - x^2}$, passing through points $(-r, 0)$, $(0, r)$, and $(r, 0)$, with domain $[-r, r]$ and range $[0, r]$.