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]