Core Concept
Reading Function Notation
The expression f(x) = 3x − 1 tells you three things at once:
f → the name of the function
(x) → x is the input variable
3x−1 → the rule: "multiply input by 3, then subtract 1"
Different letters name different functions: f, g, h, p, q... even A(r) (area as a function of radius) is perfectly valid notation.
Interpreting Notation
What Do These Mean?
Given f(x) = x² + 4, let's interpret several expressions:
f(0) → "f of 0" = (0)² + 4 = 4
f(a) → "f of a" = a² + 4
f(x+1) → "f of (x+1)" = (x+1)² + 4 = x² + 2x + 5
f(2t) → "f of 2t" = (2t)² + 4 = 4t² + 4
2·f(x) → 2 times f(x) = 2(x² + 4) = 2x² + 8
Notice the difference between f(2t) (input is 2t) and 2·f(t) (2 times the output of f at t).
Real-World Notation
Meaningful Function Names
Functions don't have to be called f. Descriptive names make math more readable:
C(n) = 5n + 20 (Cost in dollars for n items)
A(r) = πr² (Area of circle with radius r)
H(t) = −5t² + 20t (Height in metres at time t seconds)
Then H(3) means "the height at time t = 3 seconds."
Example
Solving for x Given f(x)
Given f(x) = 4x − 7, find x when f(x) = 9.
1. Set the function equal to 9:
2. 4x − 7 = 9
3. 4x = 16
4. x = 4
Check: f(4) = 4(4) − 7 = 16 − 7 = 9 ✓
Watch Out
Common Mistakes
- Reading f(x) as "f times x" — it means "f of x"
- Confusing f(2x) (substitute 2x) with 2f(x) (multiply output by 2)
- Confusing f(x+1) with f(x)+1 — these give different results!
- Parentheses in function notation mean INPUT, not multiplication
Your Turn
Try It Yourself
Let f(x) = 3x + 2 and g(x) = x².
Q1. Find f(x + 2).
Show Answer
f(x+2) = 3(x+2) + 2 = 3x + 6 + 2 = 3x + 8
Q2. Find 2·g(x) vs. g(2x). Are they different?
Show Answer
2·g(x) = 2·x² = 2x²
g(2x) = (2x)² = 4x²
Yes — they are different!
Q3. If f(x) = 6x − 3, find x when f(x) = 15.
Show Answer
6x − 3 = 15 → 6x = 18 → x = 3
Key Takeaways
1-Minute Summary
- f(x) = "f of x" — f is the name, x is the input
- To evaluate, replace x with whatever is in the parentheses
- f(2x) ≠ 2f(x) — always check what the input is
- To solve for x: set the rule equal to the given output, then solve