What Is a Function, and What Are Its Domain and Range?

A function is a relation that assigns every permitted input exactly one output. Its domain is the set of permitted input values, while its range is the set of output values the function can actually produce.

How a function works

In the notation $y=f(x)$, $x$ is the input and $f(x)$ is the corresponding output. A function may assign different inputs the same output, but one input cannot be assigned two different outputs.

For a graph, this requirement can be checked using the vertical-line test: if any vertical line intersects the graph more than once, the relation is not a function.

Consider

$$f(x)=\sqrt{x-2}+1.$$

The expression inside the square root must be non-negative:

$$x-2\geq 0 \Rightarrow x\geq 2.$$

Therefore, the domain is $x\geq 2$, or $[2,\infty)$. Since $\sqrt{x-2}\geq 0$, adding gives . Therefore, the range is , or .

On a GDC, graph the function and use the displayed window to identify possible endpoints, but justify restrictions from the formula or context. For example, a denominator cannot equal zero. Remember that the default domain is the largest real domain for which the function is defined unless the question states otherwise.

A common misconception is that the domain and range are simply all real numbers. Restrictions can arise from the formula or context: division by zero is undefined, square roots require non-negative inputs, and a real-world model may exclude negative time or non-integer quantities.

IB exam technique

This is IB Mathematics: Applications and Interpretation SL 2.2, commonly tested in a Paper 1 short-response style, although a GDC is available on every AI paper. State the domain and range clearly using inequalities, interval notation, or set notation, and distinguish values included at an endpoint from those excluded.