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.)
Drag the vertical line across each graph. If it ever meets the curve at two points at once, that single input has two outputs — so it is not a function. A parabola passes; a sideways curve fails.
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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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f(a) reads UP then ACROSS; f(x) = k reads ACROSS then DOWN: Given a graph, to find f(a) go up from a on the x-axis to the curve, then across to the y-axis.
To solve f(x) = k, go across from y = k to the curve, then down — and watch for more than one answer.
Click 'Walk me through it' for the read-off step by step, then switch to 'Practice' and read a few off yourself before revealing.
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IB-style question — read from the graph
The graph of y = f(x) for −1 ≤ x ≤ 5 is shown above.
Write down:
(a) f(2);
(b) f(0);
(c) the values of x for which f(x) = 5.
Step by step
- (a) f(2): go UP from x = 2 to the curve, then ACROSS to the y-axis.
- (b) f(0): up from x = 0, then across.
- (c) f(x) = 5: go ACROSS from y = 5 to the curve, then DOWN — it meets the curve twice.
Final answer
(a) 6 (b) 2 (c) x = 1 and x = 3.
f(x) = k can have more than one answer: Reading across often hits the curve twice — give all the x-values.
And don't mix them up: f(a) starts on the x-axis (read up); solving f(x) = k starts on the y-axis (read across).
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.
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A table is just f written out: Each column pairs an input x with its output f(x).
To read f(a), find a in the x-row and take the value directly below it.
To solve f(x) = k, scan the f(x)-row for k and read off the x above it.
A second row (like g) works exactly the same way.
| x | −1 | 0 | 2 | 5 |
|---|---|---|---|---|
| f(x) | 4 | 2 | −1 | 6 |
| g(x) | 2 | 5 | 0 | −1 |
IB-style question — from a table
The table above shows values of f(x) and g(x); both f and g are one-to-one.
Find:
(a) g(0);
(b) f(2);
(c) the value of x for which f(x) = 6.
Step by step
- (a) Read g(0) straight from the g-row, under x = 0.
- (b) Read f(2) from the f-row, under x = 2.
- (c) Scan the f-row for 6 and read the x above it.
Final answer
(a) 5 (b) −1 (c) x = 5.
Down for a value, across for an input: To read f(a) or g(a), go down the column under a.
To solve f(x) = k, go across the f-row to find k, then read the x above it.
The two questions move in different directions — don't mix them up.