What Is Function Notation (f(x))?

Function notation expresses the output of a function for a given input. In $f(x)$, $f$ names the function, $x$ represents the input, and $f(x)$ represents the corresponding output.

A function maps each permitted input to exactly one output. For example, suppose

$$f(x)=2x+3.$$

This rule tells you to multiply the input by $2$, then add $3$. To evaluate $f(4)$, substitute $x=4$:

$$f(4)=2(4)+3=11.$$

Therefore, an input of $4$ produces an output of $11$. Graphically, this means that the point $(4,11)$ lies on the graph $y=f(x)$.

Substitution must replace every occurrence of the input variable. For example, if $g(x)=x^2-4x$, then

$$g(-2)=(-2)^2-4(-2)=4+8=12.$$

The brackets around $-2$ are important because the input is negative. Function notation can also define equations: $f(x)=7$ asks which input values produce an output of $7$, whereas $f(7)$ asks for the output produced by input $7$. These are different tasks.

Function notation also connects to the domain and range. The domain is the set of permitted inputs, while the range is the set of possible outputs. For $f(x)=\frac{1}{x-2}$, the domain excludes $x=2$, because division by zero is undefined. If a proposed input is outside the domain, the function value is undefined. The range may be read from the outputs or graph.

A common misconception is that $f(x)$ means $f\times x$. It does not: the parentheses indicate that the function $f$ is being applied to the input $x$. Similarly, $f^{-1}(x)$ means the inverse function, not .

Exam technique: This is IB Mathematics AA SL 2.2, examinable on Papers 1 and 2. For a command such as find $f(3)$, show the substitution and simplification rather than writing only the final value. Always check whether the stated domain permits the input. On Paper 1, calculate analytically; on Paper 2, technology may support graph-based questions.