Function Notation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Function Notation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Function notation f(x) is a shorthand that names a function (f) and specifies its input (x). Writing f(3) = 10 means that when the input is 3, the function produces the output 10. This notation is not multiplication.

The notation f(x) is not "f times x" — it means "the output of function f when the input is x." The parentheses contain the input, not a multiplication.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: f(x) names both the function (f) and the current input (x). Replacing x with any expression gives the output for that specific input.

Common stuck point: f(x+1) \neq f(x) + 1 in general — you must substitute (x+1) for every occurrence of x in the formula, then simplify.

Sense of Study hint: To evaluate f(a), replace every x in the formula with a, then simplify step by step. For example, if f(x) = 2x + 3, then f(5) = 2(5) + 3 = 13.

Worked Examples

Example 1

easy
If f(x) = 3x^2 - 2x + 1, find f(-2).

Solution

  1. 1
    Replace every x in the formula with -2: f(-2) = 3(-2)^2 - 2(-2) + 1.
  2. 2
    Evaluate: f(-2) = 3(4) + 4 + 1 = 12 + 4 + 1.
  3. 3
    f(-2) = 17.

Answer

f(-2) = 17
Function notation f(x) names the function (f) and its input variable (x). To evaluate f(a), substitute a for every occurrence of x in the expression. Be careful with signs when substituting negative values — use parentheses around the substituted value.

Example 2

medium
If g(x) = x^2 + 3x, find and simplify \frac{g(x+h) - g(x)}{h}.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
If f(x) = 2x - 1 and g(x) = x^2 + 3, find (f \circ g)(x) and (g \circ f)(x).

Example 2

hard
A function satisfies f(x+1) = 2f(x) + 3 for all x, with f(0) = 1. Find f(1), f(2), and f(3).
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Background Knowledge

These ideas may be useful before you work through the harder examples.

function definitionvariablesevaluation