Domain and Range: The Inputs and Outputs of a Function

Look for denominators or square roots. If there are none, nothing is excluded.

Domain: all real numbers. Range: all real numbers.

No denominator and no root, so every input works and the domain is all real numbers.

Any output $y$ is reached by the input $x = y - 6$, so the range is all real numbers too.

Domain: all real numbers. Range: $\{-2\}$.

Every input is allowed, but the output never changes, so the range is the single value $-2$.

Which input makes the denominator zero?

All real numbers except $-2$.

Set $x + 2 = 0$, giving $x = -2$. That input would produce $\tfrac{4}{0}$, which is undefined, so it is excluded.

$x \ge -1$

The expression under the root must be at or above zero: $x + 1 \ge 0$, so $x \ge -1$.

Domain: $\{2, 4, 6\}$. Range: $\{5, 9\}$.

The inputs are $2$, $4$ and $6$. The outputs are $5$, $5$ and $9$, and since a range is a set, the repeated $5$ is written once.

A segment stops at its endpoints.

Domain: $0 \le x \le 4$. Range: $3 \le y \le 11$.

The segment spreads from $x = 0$ to $x = 4$, and from $y = 3$ to $y = 11$. Both endpoints are included, so both use $\le$.

Factor the denominator before setting it to zero.

All real numbers except $3$ and $-3$.

Set $x^2 - 9 = 0$. Factoring as a difference of squares gives $(x - 3)(x + 3) = 0$, so $x = 3$ or $x = -3$.

Both inputs make the denominator zero, so both are excluded.

Think about what the inputs and outputs count, and whether fractions make sense for each.

Domain: whole numbers $0, 1, 2, \ldots$. Range in practice: whole numbers, after rounding up.

The input counts students, so it must be a whole number and cannot be negative: the domain is $\{0, 1, 2, \ldots\}$.

The formula’s outputs are often fractional — $20$ students gives $\tfrac{20}{6} \approx 3.33$ pizzas — but you cannot order a third of a pizza. In practice you round up to $4$, so the useful range is also whole numbers.

This is a case where the mathematics and the situation disagree, and the situation wins.