Function Notation
f(x) is a name for an output, not multiplication.
The explanation
f(x) is read "f of x." It does not mean f times x.
f(x) = 2x + 1 is the same rule as y = 2x + 1, but the notation lets you name the input you are using.
f(3) means substitute 3: f(3) = 2(3) + 1 = 7. The pair (3, 7) is on the graph.
You can substitute anything, including expressions. f(a + 1) = 2(a + 1) + 1 = 2a + 3.
Solving f(x) = 11 is the reverse question: which input gives 11? Set 2x + 1 = 11 and solve, giving x = 5.
The notation names a function and displays its argument, so f(3) denotes the image of 3 under f — a number, not a product. The variable inside is a bound placeholder: f(x) = 2x + 1 and f(t) = 2t + 1 define the same function.
Substituting an expression is composition in miniature, and requires parentheses around the substituted expression to preserve grouping. Common evaluations worth being fluent in: f(−x) (used for even/odd symmetry), f(x + h) (used in the difference quotient (f(x+h) − f(x))/h, the foundation of the derivative), and f(g(x)).
Two distinct questions share the notation and are often confused. Evaluating f(a) is direct substitution with a unique answer. Solving f(x) = b asks for every input mapping to b, and may have none, one or many solutions. The first traverses the function forwards, the second backwards.
Worked example
For f(x) = x² − 4x, find f(−2), and solve f(x) = 0.
- f(−2) = (−2)² − 4(−2) = 4 + 8 = 12.
- Solve x² − 4x = 0 by factoring: x(x − 4) = 0.
- So x = 0 or x = 4.
Answer: f(−2) = 12; f(x) = 0 at x = 0 and x = 4
Common mistakes
- Reading f(3) as f times 3.
- Forgetting parentheses when substituting a negative, turning f(−2) into −2² − 4(−2).