Mathwords logoReference LibraryMathwords

Function Notation

Function notation is a way of writing functions using a name (like ff) and an input variable in parentheses, such as f(x)f(x). It tells you both the name of the function and what variable goes in, so f(x)=2x+3f(x) = 2x + 3 means "the function ff takes an input xx and gives back 2x+32x + 3."

Function notation expresses a function by assigning it a name, typically ff, gg, or hh, followed by its input variable enclosed in parentheses. The expression f(x)f(x) is read as "ff of xx" and represents the output value of the function ff when the input is xx. This notation makes it easy to refer to specific outputs — writing f(4)f(4) means "substitute 4 for xx and compute the result." Functions can use any letter as a name, and xx can be replaced by numbers, other variables, or even entire expressions.

Key Formula

f(x)=expression in terms of xf(x) = \text{expression in terms of } x
Where:
  • ff = the name of the function
  • xx = the input variable (also called the argument)
  • f(x)f(x) = the output value for input x

Worked Example

Problem: Given f(x)=3x−5f(x) = 3x - 5, find f(4)f(4) and f(−2)f(-2).
Step 1: To find f(4)f(4), replace every xx in the expression with 4.
f(4)=3(4)−5f(4) = 3(4) - 5
Step 2: Simplify the arithmetic.
f(4)=12−5=7f(4) = 12 - 5 = 7
Step 3: To find f(−2)f(-2), substitute −2-2 for xx.
f(−2)=3(−2)−5f(-2) = 3(-2) - 5
Step 4: Simplify.
f(−2)=−6−5=−11f(-2) = -6 - 5 = -11
Answer: f(4)=7f(4) = 7 and f(−2)=−11f(-2) = -11.

Visualization

Why It Matters

Function notation is the standard language used across algebra, calculus, science, and engineering to describe how quantities depend on each other. When a physicist writes v(t)v(t) for velocity as a function of time, or a programmer defines a function that converts temperatures, they are using this same notation. Mastering it early makes reading and writing mathematical relationships far more efficient.

Common Mistakes

Mistake: Interpreting f(x)f(x) as ff multiplied by xx.
Correction: The parentheses in function notation do not mean multiplication. f(x)f(x) means "the output of function ff when the input is xx." If you need multiplication, you would write it differently, such as f⋅xf \cdot x.
Mistake: Forgetting to substitute into every instance of xx in the expression.
Correction: When evaluating f(3)f(3) for f(x)=x2−2x+1f(x) = x^2 - 2x + 1, you must replace both the x2x^2 and the −2x-2x with 3: f(3)=(3)2−2(3)+1=4f(3) = (3)^2 - 2(3) + 1 = 4.

Related Terms

  • Function — The concept that function notation represents
  • Evaluate — Finding an output by substituting an input
  • Domain — The set of all valid inputs for a function
  • Range — The set of all possible outputs of a function