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.

Formulas and Graphs are tools representing those pairs.

In fact, there are many formulas representing the same function.

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

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

Lerning Outcomes

In this section, students will

• compare equivalent forms.
• compare nearly equivalent forms.