Function Notation
Master f(x) notation, learn to evaluate functions at specific values and expressions, and explore function operations.
What is Function Notation?
Instead of writing y = 2x + 3, we use function notation: f(x) = 2x + 3. This is read as "f of x equals 2x + 3".
Function name
f
Input variable
(x)
Rule / Formula
= 2x + 3
The notation f(x) does not mean "f times x". It means "the output of function f when the input is x". Different letters can name different functions, such as g(x), h(x), or P(x).
A key advantage of function notation is that we can clearly specify the input. For example, f(3) means "evaluate f at x = 3", giving f(3) = 2(3) + 3 = 9.
Evaluating Functions
To evaluate a function at a given value, substitute that value everywhere you see x in the formula.
Types of Evaluation
At a number: f(5)
Replace every x with 5 and simplify.
At an expression: f(a + 1)
Replace every x with (a + 1), using brackets to preserve order of operations.
At another function: f(g(x))
This is a composite function — replace x in f with the entire expression g(x).
Operations on Functions
Functions can be combined using arithmetic operations:
| Operation | Notation | Definition |
|---|---|---|
| Sum | (f + g)(x) | f(x) + g(x) |
| Difference | (f - g)(x) | f(x) - g(x) |
| Product | (f · g)(x) | f(x) × g(x) |
| Quotient | (f/g)(x) | f(x) / g(x), where g(x) ≠ 0 |
Key Vocabulary
Function
A rule that assigns exactly one output to each input. Each x-value maps to exactly one y-value.
f(x)
Read "f of x". Represents the output of function f when the input is x. Not multiplication.
Evaluate
To substitute a value into the function and calculate the result.
Independent Variable
The input variable (usually x) whose value is freely chosen.
Worked Examples
If f(x) = 3x² - 2x + 1, find f(-2).
Step 1: Replace every x with (-2): f(-2) = 3(-2)² - 2(-2) + 1.
Step 2: Calculate: = 3(4) + 4 + 1 = 12 + 4 + 1.
Answer: f(-2) = 17.
If f(x) = x² + 1, find f(a + 3).
Step 1: Replace x with (a + 3): f(a + 3) = (a + 3)² + 1.
Step 2: Expand: = a² + 6a + 9 + 1.
Answer: f(a + 3) = a² + 6a + 10.
If f(x) = 2x + 1 and g(x) = x², find (f + g)(3) and (f · g)(2).
(f + g)(3): f(3) + g(3) = (2(3) + 1) + (3²) = 7 + 9 = 16.
(f · g)(2): f(2) × g(2) = (2(2) + 1) × (2²) = 5 × 4 = 20.
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
If f(x) = 4x - 7, what is f(3)?
Question 2
If g(x) = x² - 4x, what is g(-1)?
Question 3
If f(x) = 2x + 1, which expression represents f(x + h)?
Question 4
If f(x) = x + 5 and g(x) = 3x, what is (f · g)(2)?
Question 5
If f(x) = x² and f(k) = 49, what is k (where k > 0)?
Key Concepts Summary
- ●f(x) means "the output of function f when x is the input" — it is not multiplication.
- ●To evaluate f(a), replace every x with a in the formula and simplify.
- ●When substituting an expression, use brackets to maintain correct order of operations.
- ●Functions can be combined using addition, subtraction, multiplication, and division.
- ●A function assigns exactly one output to each input in its domain.