Summary of Review of Functions

Essential Concepts

  • 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]

Glossary

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]
dependent variable
the output variable for a function
domain
the set of inputs for a function
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]
independent variable
the input variable for a function
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]
range
the set of outputs for a function
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]