A function is a machine: A function f is a rule: put a number in, get exactly one number out. f(x) is the output for input x — so f(3) means "put 3 into the rule."
IB-style question — read the machine
For f(x) = 2x + 1, find f(3) and f(−2).
Step by step
- f(3): replace every x with 3.
- f(−2): replace every x with −2 (use brackets).
Final answer
f(3) = 7 and f(−2) = −3.
f(x) is NOT f times x: f(x) is read "f of x" — the function applied to x. The brackets hold the input, they are not multiplication.
One input, one output: For something to be a function, each input may give only one output. (Different inputs can share an output — that's allowed.)
Replace every x, then simplify: To find f(a), write a in place of every x — wrapping it in brackets so signs and powers behave — then simplify.
IB-style question — a negative input
For g(x) = x² − 4x, find g(−3).
Step by step
- Substitute x = −3 in brackets.
- Square and multiply.
- Add.
Final answer
g(−3) = 21.
IB-style question — an algebraic input
For f(x) = 3x − 5, find f(2a).
Step by step
- Replace x with the whole expression 2a.
- Simplify.
Final answer
f(2a) = 6a − 5.
Brackets save you: Without brackets, (−3)² becomes −9 by mistake. Always write (−3)²= 9. The same care applies when the input is an expression.
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Given the output, find the input: Evaluating goes input → output. Solving f(x) = k goes the other way: set the rule equal to k and solve for x.
IB-style question — a linear rule
For f(x) = 2x + 1, solve f(x) = 9.
Step by step
- Set the rule equal to 9.
- Solve.
Final answer
x = 4.
IB-style question — two inputs, one output
For f(x) = x² − 3, solve f(x) = 6.
Step by step
- Set equal to 6.
- Solve — remember both roots.
Final answer
x = 3 or x = −3 (two inputs give the same output).
Don't lose a solution: Quadratics (and other curves) can send two inputs to the same output, so f(x) = k may have more than one answer — write them all.
f(a) reads up; f(x) = k reads across: On a graph, f(a) = the y-value when x = a — go up from a on the x-axis to the curve, then across to the y-axis. To solve f(x) = k, read across from y = k to the curve, then down to the x-axis.
IB-style question — from a table
A function is given by the table f(1) = 4, f(2) = 7, f(3) = 4, f(4) = 1.
Write down f(3), and find every x with f(x) = 4.
Step by step
- Read f(3) straight from the table.
- Find every input whose output is 4.
Final answer
f(3) = 4; and f(x) = 4 at x = 1 and x = 3.
An output can repeat: Here two different inputs (1 and 3) give the same output 4 — perfectly fine for a function. But each single input still has just one output.