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Function notation

Read f(x) as the output of a function at an input. Evaluate a rule carefully, check its domain, and connect notation to a table or graph.

Fraction pieces and simple shapes illustrate relationships between numbers.

What you’ll learn

  • Explain what f(x) represents.
  • Substitute an input into every occurrence of the variable.
  • Check whether the input belongs to the function’s domain.

The notation names an output

In f(x), f names the function and x identifies the input. The notation does not mean f multiplied by x. Different functions can produce different outputs from the same input.

Substitute into the whole rule

Replace every occurrence of the input variable and preserve parentheses and operation order. Also check the domain; division by zero or a stated restriction can exclude a proposed input.

A mistake to avoid

A function can give the same output for two different inputs. That does not violate the function rule: what matters is that each allowed input has one output.

Worked example

Evaluate f(−2) when f(x) = x² − 3

Substitute −2 in parentheses: f(−2) = (−2)² − 3 = 4 − 3 = 1. Here f(2) also equals 1. Both inputs are allowed and each has a single output.

Learning paths that include this skill

Choose the route that fits your goal. The same skill can be useful in more than one subject or stage of learning.