Definition, Domain, Range

What is a Function?

A function is a rule that takes an input and produces exactly one output.

Think of it like a machine: you put something in, something comes out.

The key rule: one input, one output.

You can have different inputs giving the same output (like 3 and -3 both giving 9 when squared).

But you cannot have one input giving multiple outputs.

Function vs. Not a Function

In Relation A, each input maps to exactly one output. That’s a function.

In Relation B, input 1 maps to both a and b. That violates the rule.

A function gives each input exactly one output.

Function Notation

We write $f(x)$ to mean “the output of function $f$ when the input is $x$.”

$f(x) = x^2$

This says: “take the input, square it.”

• $f(3) = 3^2 = 9$
• $f(-2) = (-2)^2 = 4$
• $f(a+1) = (a+1)^2$

The letter doesn’t have to be $f$. You’ll also see $g(x)$, $h(x)$, or even $P(t)$ for population as a function of time.

The variable doesn’t have to be $x$ either. It’s just a placeholder for “the input.”

Domain

The domain is the set of all inputs that work.

For $f(x) = x^2$, any real number works.

Domain = all real numbers

But some functions have restrictions. There are three common ones:

1. Division by zero

$f(x) = \frac{1}{x}$

Can’t divide by zero, so $x \neq 0$.

Domain: all reals except 0

2. Square roots of negatives

$f(x) = \sqrt{x}$

Can’t take the square root of a negative (in real numbers).

Domain: $x \geq 0$

3. Logarithms of non-positives

$f(x) = \log(x)$

Logarithms only work for positive numbers.

Domain: $x > 0$

Finding Domain

When given a function, ask yourself:

1. Are there any denominators? → Set them $\neq 0$
2. Are there square roots? → Set the inside $\geq 0$
3. Are there logarithms? → Set the argument $> 0$

Example: Find the domain of $f(x) = \dfrac{\sqrt{x-3}}{x-5}$

Two restrictions:

• Square root: $x - 3 \geq 0 \implies x \geq 3$
• Denominator: $x - 5 \neq 0 \implies x \neq 5$

Combine them: $x \geq 3$ and $x \neq 5$

Domain: $[3, 5) \cup (5, \infty)$

Range

The range is the set of all possible outputs.

For $f(x) = x^2$:

• Squaring always gives a non-negative result
• The smallest output is 0 (when $x = 0$)
• There’s no largest output

Range: $[0, \infty)$

Another example: $f(x) = \dfrac{1}{x}$

• As $x$ gets very large, $\frac{1}{x}$ approaches 0 (but never reaches it)
• As $x$ gets very small (negative), same thing
• The output can be any number except 0

Range: all reals except 0