Range

Definition

The range of a function is the set of actual outputs it produces.

Example:

f(x) = x + 1
Domain: {1, 2, 3}

Outputs:

f(1) = 2
f(2) = 3
f(3) = 4

So the range is:

{2, 3, 4}

Range = outputs that actually occur.

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Range In Ordered Pairs

For a function represented as:

f = {(1, a), (2, b), (3, b)}

the range is:

{a, b}

Even though b appears more than once, it is listed once in the range because sets do not duplicate elements.

In ordered pairs, the range comes from the output positions.

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Domain, Codomain, Range

These three ideas answer different questions.

Domain: what inputs are allowed?
Codomain: where outputs are allowed to land?
Range: what outputs actually happen?

Example:

f(x) = x * x
Domain: {-2, -1, 0, 1, 2}
Codomain: {0, 1, 2, 3, 4}
Range: {0, 1, 4}

The function never outputs 2 or 3.

Range is determined by applying the function to the domain.

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Why Range Matters

The range tells us what a function actually reaches.

This matters when asking:

• which outputs are possible?
• which outputs are impossible?
• does the function cover its codomain?

Range describes the real behavior of the function.

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Final Summary

• The range is the set of actual outputs.
• The range comes from applying the function to its domain.
• In ordered pairs, the range is the set of second components that occur.
• The range can be smaller than the codomain.
• Domain, codomain, and range describe different parts of a function.