Equal Functions

Two functions that look algebraically identical may not be equal if their domains differ. This section defines function equality precisely and uses examples to show why domain is an essential part of a function's identity.

Quick Reference

Condition for	Requirement
Same domain
Same outputs	for every in the common domain
Consequence	Graphs of and are identical

When Are Two Functions Equal?

Consider and .

Algebraically, when . But , while is undefined (division by zero). So but . Their domains differ, and their graphs differ: the graph of has a hole at that the graph of does not.

Graph of f(x) = x, the line through the origin. — (a) Graph of .

Graph of g(x) = x²/x, the same line but with a hole at the origin. — (b) Graph of . There is a hole at because is undefined there.

Two functions and are equal (or identical) if and only if:

They have the same domain: .
for every in their domains.

If and are equal, their graphs are identical and their ranges are equal: .

Examples

Consider and . Are and equal?

Solution

Simplify

for all

. However, the domains differ:

Since their domains are not equal,

and

are not equal functions.

Let and . Are and equal?

Solution

For all real

Both functions have domain

and produce identical outputs. Therefore,

and

are equal.

Let and . Are and equal?

Solution

Simplify

Both simplify to

, and crucially, their domains are identical: both are undefined only at

Since the domains match and

for all

, the functions

and

are equal.

Frequently Asked Questions

If two formulas simplify to the same expression, are the functions equal?

Not necessarily. They are equal only if the domains also match. Simplifying can change the domain. For example, canceling a factor that is zero at some point removes that point from the domain, making the simplified function different from the original.

Can two functions be equal but their formulas look completely different?

Yes. For example,

and

look different but are equal because they have the same domain and the same output for every input.

Does equal domain guarantee equal functions?

No. Equal domain is necessary but not sufficient. The functions must also produce the same output for every input in that domain.

What is the relationship between equal functions and their graphs?

Two functions are equal if and only if their graphs are identical. A hole, break, or any difference in a single point makes the graphs (and therefore the functions) unequal.

Quick Reference

When Are Two Functions Equal?

Examples

Frequently Asked Questions

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