To evaluate a function, replace every x in the rule with the input, writing the input inside brackets, then work out the result in the usual order. The brackets are what keep a negative sign attached to its number.
This skill is the entry point to functions and mappings. It returns in composite functions, inverse functions, graphs and solving equations, so a sign slip here travels a long way.
What does a function rule do?
A function is a rule that takes an input and gives exactly one output. We write f(x) = 2x + 3, or f: x → 2x + 3, and both mean the same thing: “double the input, then add 3”.
The letter x is a placeholder. When the question says f(−4), it is telling you what goes into the placeholder.
How to evaluate, step by step
- Write the rule with empty brackets where x was: f( ) = ( )² − 4( ) + 1.
- Put the input inside every pair of brackets, sign included.
- Do powers first, then multiplication, then addition and subtraction.
- Watch double signs. Subtracting a negative, such as − 4(−3), becomes adding.
- Write the answer as f(−3) = 22, so the input and output stay linked.
Worked example
If f(x) = x² − 4x + 1, find f(−3) and f(−1).
Step 1, brackets: f(−3) = (−3)² − 4(−3) + 1.
Step 2, power: (−3)² = 9, because −3 × −3 = 9.
Step 3, multiplication: −4 × (−3) = +12.
Step 4, add up: 9 + 12 + 1 = 22.
So f(−3) = 22.
For f(−1): (−1)² − 4(−1) + 1 = 1 + 4 + 1 = 6.
Check: try f(0) = 0 − 0 + 1 = 1. The outputs 22, 6, 1 fall steadily as the input rises from −3 to 0, which matches a curve sloping down on the left side of its lowest point. A sudden jump would be a warning.
The mistake to watch for
The usual slip is to lose the brackets when squaring.
Mistaken working: f(−3) = −3² − 4(−3) + 1 = −9 + 12 + 1 = 4
The student wrote −3² and treated it as “square 3, then make it negative”.
The correct reading is that x is replaced by the whole of −3, so the square is (−3)² = 9, giving 22, not 4. The fix is mechanical: put every substituted input in brackets before you calculate anything, and the problem disappears.
Check yourself
Work on paper first, then open each answer.
1. If g(x) = 2x² − 3x, find g(−2).
Show answer
g(−2) = 2(−2)² − 3(−2) = 2(4) + 6 = 8 + 6 = 14.
2. If h(x) = 5 − x², find h(−4).
Show answer
h(−4) = 5 − (−4)² = 5 − 16 = −11.
3. If f(x) = 3 − 2x, find f(−5), then find the input that gives an output of 3.
Show answer
f(−5) = 3 − 2(−5) = 3 + 10 = 13. For an output of 3: 3 − 2x = 3, so x = 0.
Where this leads next
Once negative inputs feel safe, try tracing an input through a composite function, where one output becomes the next input. Then test yourself with the functions and mappings practice set. The function composition and inverse explorer lets you check a substitution instantly, and the non-calculator working trainer builds the habit of showing each step.
Some students can follow this in class but lose the sign when a question is longer or timed. That is the kind of pattern our teachers look for in online one-to-one Mathematics tuition.