Functions, a review for calculus

What a function is

In calculus we work with functions whose inputs and outputs are real numbers, usually given by a formula. The graph of \(f\) is the set of points \((x, f(x))\). The vertical line test checks whether a graph is a function: no vertical line may cross it more than once.

Domain and range

The domain is the set of allowed inputs; the range is the set of outputs the function actually produces. For a formula, the domain is every real number except where something breaks:

Even and odd functions

Most functions are neither.

Increasing and decreasing

\(f\) is increasing on an interval if larger inputs give larger outputs there, and decreasing if larger inputs give smaller outputs. Derivatives will give us a precise way to find these intervals.

Composition

The chain rule for derivatives is all about compositions, so it pays to spot the "inside" and "outside" functions quickly.

Inverse functions

A one-to-one function (different inputs always give different outputs) has an inverse \(f^{-1}\) that undoes it: \(f^{-1}(f(x)) = x\). The graph of \(f^{-1}\) is the reflection of the graph of \(f\) across \(y = x\).

Practice