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 $y=f(x)$ , the domain is considered to be the set of all real numbers $x$ for which the function is defined.
When sketching the graph of a function $f$ , each vertical line may intersect the graph, at most, once.
A function may have any number of zeros, but it has, at most, one $y$ -intercept.
To define the composition $g\circ f$ , the range of $f$ must be contained in the domain of $g$ .
Even functions are symmetric about the $y$ -axis whereas odd functions are symmetric about the origin.

Key Equations

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

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

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

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

even function: a function is even if $f(−x)=f(x)$ for all $x$ in the domain of $f$

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 $(x,y)$ such that $x$ is in the domain of $f$ and $y=f(x)$

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

odd function: a function is odd if $f(−x)=−f(x)$ for all $x$ in the domain of $f$

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

symmetry about the $y$ -axis: the graph of a function $f$ is symmetric about the $y$ -axis if $(−x,y)$ is on the graph of $f$ whenever $(x,y)$ 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 $x$ is a zero of a function $f$ , $f(x)=0$