Formulas are not functions. Graphs are not functions.

A function is a collection of three sets: a domain, a range, and a set of pairs, along with one rule, which states that each domain item is included in exactly one pair.

Functions do not have Variables

Formulas and Graphs are tools representing those pairs. They have variables, which we use to represent pieces of the function.

In fact, there are always many formulas representing the same function. This is very very very helpful.

For example, each formula below represents the same function on \((-\infty , \infty )\).

• \(x^2\)
• \((x - 1)^2 + 2 x - 1\)
• \(\sqrt {x^4}\)
• \( x \cdot x\)
• \( 4 \cdot \left ( \frac {x}{2} \right )^2\)
• \(\begin{cases} \frac {x^3}{x} &\text {if $x \ne 0$,}\\ 0 &\text {if $x = 0$}. \end{cases}\)

Much of our algebraic reasoning relies on stepping through equivalent formulas.

Lerning Outcomes

In this section, students will

• compare equivalent forms.
• compare nearly equivalent forms.