What is function notation?

Function notation is a way of writing functions clearly so you know:

• The function’s name

• What the input is

• What the output is

Instead of writing y = 2x + 3, we can write:

• f = name of the function (can also be g, h, etc.)

• x = input

• f(x) = output

Read it as: “f of x equals 2x plus 3.”

How function notation works

Step 1: Identify the input

• The input is the value you put into the function.

• Example: if f(x) = 2x + 3, input could be x = 4.

Step 2: Substitute the input

• ✅ Output is 11

Step 3: Write the output as f(input)

• This makes it clear that 11 is the output of the function when the input is 4.

Evaluating functions

Example 1:

• Find f(3)

✅ Output is 4

Example 2:

• Find g(-2)

✅ Output is -3

Using function notation with multiple functions

• You can have different function names: f(x), g(x), h(x).

• Each function has its own rule.

Example:

• Find f(3) → 3 + 2 = 5

• Find g(3) → 3 × 3 = 9

Function notation vs. y = ?

• y = 2x + 3 and f(x) = 2x + 3 mean the same thing.

• Function notation is more precise and works well when:

  • You have multiple functions

  • You want to substitute expressions instead of just numbers

Example:

Domain and range in function notation

• Domain = all possible inputs for x

• Range = all possible outputs for f(x)

Example:

• Domain: all real numbers (−∞ < x < ∞)

• Range: all real numbers ≥ 0 (f(x) ≥ 0)

Common beginner mistakes

1. ❌ Confusing f(x) with f × x (f(x) is not multiplication)

2. ❌ Forgetting to substitute the input correctly

3. ❌ Mixing up input (x) and output (f(x))

4. ❌ Not using parentheses when substituting expressions

Why function notation matters

• Makes algebra clearer and more organized

• Essential for advanced algebra topics: compositions, inverses, transformations

• Shows clearly input → function → output