Precalculus

Functions

• definition. A relation ${f}$ from a set ${A}$ to a set ${B}$ is a function if, for each ${x \in A,}$ there is at most one element ${y \in B.}$
  • Functions are sometimes called mappings.
• We can denote a function with the notation: $$f: A \to B.$$
  • The domain of ${f}$ is the set ${A}$ or a subset of ${A.}$
    • The codomain of ${f}$ is the set ${B.}$
    • The range of ${f}$ is a subset of ${B.}$
• example. Below are relations ${f,}$ ${g,}$ and ${h.}$

  

  • ${f}$ is not a function since one input maps to two elements in the codomain.
  • ${g}$ is a function since each input maps to at most one element in the codomain.
  • ${h}$ is a function since each input maps to at most one element in the codomain. Notice that not every input must be mapped. The only requirement is that an input maps to at most one.

Affine Functions

• definition. An affine function is a function defined by an equation of the form $$f(x) = ax + b,$$ where ${a}$ and ${b}$ are constants in ${\reals}$.
  • If ${b = 0}$ and ${a \neq 0,}$ we call ${f}$ a linear function.
• example. ${f(x) = 2x + 1}$ is an affine function.
• example. ${f(x) = 2x}$ is a linear function.
• example. ${f(x) = {x}/{3}}$ is a linear function.
• example. ${f(x) = \dfrac{x}{3} + \pi}$ is an affine function.
• example. Below are graphs of affine functions.

• theorem. If ${f}$ is an affine function then it has a constant rate of change equal to the slope of its graph.
• theorem. If a function ${f}$ has a constant rate of change, then it must be an affine function.

Quadratic Functions

• definition. A quadratic function is a function of the form $$f(x) = ax^2 + bx + c,$$ where ${a,b,c}$ are real constants and ${a \neq 0.}$
• The graph of a quadratic function is called a parabola.
• example. Below is the graph of ${y = 2x^2 - 2x - 1.}$