Summary of Review of Functions

A function is a mapping from a set of inputs to a set of outputs with exactly one output for each input.
If no domain is stated for a function [latex]y=f(x)[/latex], the domain is considered to be the set of all real numbers [latex]x[/latex] for which the function is defined.
When sketching the graph of a function [latex]f[/latex], each vertical line may intersect the graph, at most, once.
A function may have any number of zeros, but it has, at most, one [latex]y[/latex]-intercept.
To define the composition [latex]g\circ f[/latex], the range of [latex]f[/latex] must be contained in the domain of [latex]g[/latex].
Even functions are symmetric about the [latex]y[/latex]-axis whereas odd functions are symmetric about the origin.

Key Equations

Composition of two functions
[latex](g\circ f)(x)=g(f(x))[/latex]
Absolute value function
[latex]f(x) = |x| = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases}[/latex]

absolute value function: [latex]f(x) = |x| = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases}[/latex]

composite function: given two functions [latex]f[/latex] and [latex]g[/latex], a new function, denoted [latex]g\circ f[/latex], such that [latex](g\circ f)(x)=g(f(x))[/latex]

decreasing on the interval [latex]I[/latex]: a function decreasing on the interval [latex]I[/latex] if, for all [latex]x_1, \, x_2\in I, \, f(x_1)\ge f(x_2)[/latex] if [latex]x_1<x_2[/latex]

even function: a function is even if [latex]f(−x)=f(x)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex]

function: a set of inputs, a set of outputs, and a rule for mapping each input to exactly one output

graph of a function: the set of points [latex](x,y)[/latex] such that [latex]x[/latex] is in the domain of [latex]f[/latex] and [latex]y=f(x)[/latex]

increasing on the interval [latex]I[/latex]: a function increasing on the interval [latex]I[/latex] if for all [latex]x_1, \, x_2\in I, \, f(x_1)\le f(x_2)[/latex] if [latex]x_1<x_2[/latex]

odd function: a function is odd if [latex]f(−x)=−f(x)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex]

symmetry about the origin: the graph of a function [latex]f[/latex] is symmetric about the origin if [latex](−x,−y)[/latex] is on the graph of [latex]f[/latex] whenever [latex](x,y)[/latex] is on the graph

symmetry about the [latex]y[/latex]-axis: the graph of a function [latex]f[/latex] is symmetric about the [latex]y[/latex]-axis if [latex](−x,y)[/latex] is on the graph of [latex]f[/latex] whenever [latex](x,y)[/latex] is on the graph

table of values: a table containing a list of inputs and their corresponding outputs

vertical line test: given the graph of a function, every vertical line intersects the graph, at most, once

zeros of a function: when a real number [latex]x[/latex] is a zero of a function [latex]f[/latex], [latex]f(x)=0[/latex]