Functions
Function notation f(x), evaluating functions, and composite and inverse functions.
Learn
Functions, explained point by point
Everything the GCSE specification expects you to be able to do, and how to actually do it - the same lesson a signed-in student studies from.
Use function notation f(x) and evaluate functions
f(x) = 3x − 2 is a machine: feed in x, get out 3x − 2. Evaluating means substituting: f(4) = 10, and f(−1) = −5. The input can be an expression too - f(2a) = 6a − 2.
Solve equations of the form f(x) = k
Set the function's rule equal to the target and solve. If f(x) = 3x − 2, then f(x) = 13 means 3x − 2 = 13, so x = 5. You are answering "what input gives this output?" - the reverse of evaluating.
Find composite functions like fg(x)
fg(x) means "do g FIRST, then f" - work from the letter nearest the x outwards. With f(x) = x² and g(x) = x + 3: fg(x) = (x + 3)², but gf(x) = x² + 3. Order matters, and the two are rarely equal.
Find inverse functions f⁻¹(x)
The inverse undoes the function. Write y = f(x), swap x and y, and rearrange for y: from y = 3x − 2, swapping gives x = 3y − 2, so f⁻¹(x) = . Check: feeding f(4) = 10 into the inverse returns 4.
Practice
Try a Functions question
A GCSE-style original question from this topic. Have a go before you open the working - deciding on an answer first is what makes the working stick.
f(x) = 2x + 1 and g(x) = x − 3. Work out fg(5).
- 9
- 5
- 8
- 4
Show the answer and the working
Answer: 5
fg(5) means apply g first: g(5) = 2, then f(2) = 5.
- fg(5) means do g first, then f.
- g(5) = 5 − 3 = 2.
- f(2) = 2 × 2 + 1 = 4 + 1 = 5.
- So fg(5) = 5.