Domain

Definition

The domain of a function is the set of allowed inputs.

Example:

f(x) = x + 1
Domain: integers

This means f accepts inputs like:

-2, -1, 0, 1, 2

Domain = what the function is allowed to receive.

---

Domain Controls Valid Use

A function is only defined for inputs in its domain.

Example:

f(x) = first letter of x
Domain: non-empty words

Valid:

f("cat") = "c"

Invalid:

f(5)

The input 5 is not a word.

The domain tells us when applying a function makes sense.

---

Same Rule, Different Domain

The same rule can behave differently depending on the domain.

Example:

f(x) = x * x

If the domain is:

{1, 2, 3}

then the outputs are:

1, 4, 9

If the domain is:

{-3, -2, -1, 0, 1, 2, 3}

then the outputs include repeated results:

9, 4, 1, 0, 1, 4, 9

A function is not fully specified by the formula alone; the domain matters.

---

Domain In Ordered Pairs

For a function represented as ordered pairs:

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

the domain is:

{1, 2, 3}

It is the set of first components.

In ordered pairs, the domain comes from the input positions.

---

Final Summary

• The domain is the set of allowed inputs.
• A function is only defined on its domain.
• The same rule can behave differently with a different domain.
• In ordered pairs, the domain is the set of first components.
• Knowing the domain prevents invalid function use.