Mathematics 0580 · Syllabus tracker

Functions

Function notation, solving f(x) = k, inverse and composite functions, domain, range and asymptotes.

8 "I can…" statements 4 subtopicsIGCSE: whole chapter is Extended tier

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Quick revision

Functions on one sheet

For the night before. The full notes teach the topic; this is just what you must remember.

At a glance

Function notation, inverses, composite functions, domain, range and asymptotes.

Function notation
Inverse functions
Composite functions
Domain, range and asymptotes

1Function notation

f(x) = 3x − 4 is a rule: f(2) means put 2 in place of x: f(2) = 2.
Solve f(x) = 11 by writing 3x − 4 = 11.
f(x + 1) means replace x by (x + 1): 3(x + 1) − 4 = 3x − 1.
2f(x) = 2(3x − 4) = 6x − 8 but f(2x) = 6x − 4.

Worked example

g(x) = x² − 2x. Find g(−3) and solve g(x) = 8.

g(−3) = 9 + 6 = 15.
x² − 2x − 8 = 0 → (x − 4)(x + 2) = 0.

g(−3) = 15; x = 4 or x = −2

2Inverse functions

f⁻¹ undoes f: if f(3) = 11 then f⁻¹(11) = 3.
f⁻¹(x) is not 1/f(x).

How to do it

Write y = f(x).
Swap x and y.
Make y the subject.
Write f⁻¹(x) = …

Worked example

f(x) = (2x − 5)/3. Find f⁻¹(x).

y = (2x − 5)/3; swap: x = (2y − 5)/3.
3x = 2y − 5.
2y = 3x + 5.

f⁻¹(x) = (3x + 5)/2

Exam tip: Check: f⁻¹(f(x)) must give x back. Try a number.

3Composite functions

fg(x) means f(g(x)): do g first, then f.
Replace the x in f with the whole expression g(x), in brackets.
ff(x) means apply f twice. fg and gf are usually different.

Worked example

f(x) = 2x + 1 and g(x) = x². Find fg(x), gf(x) and fg(3).

fg(x) = f(x²) = 2x² + 1.
gf(x) = g(2x + 1) = (2x + 1)² = 4x² + 4x + 1.
fg(3) = 2 × 9 + 1 = 19.

fg(x) = 2x² + 1, gf(x) = 4x² + 4x + 1, fg(3) = 19

4Domain, range and asymptotes

Domain: the x values allowed. Leave out any value that makes a denominator zero.
Range: the y values you get out. x² is never negative, so the range of f(x) = x² is f(x) ⩾ 0.
y = k/(x − a) + b has asymptotes x = a (vertical) and y = b (horizontal).
y = a^x + b has the horizontal asymptote y = b.

Worked example

f(x) = 3/(x − 2) + 1. State the value excluded from the domain and the asymptotes.

x − 2 = 0 when x = 2.
As x gets large, 3/(x − 2) → 0, so y → 1.

x = 2 excluded; asymptotes x = 2 and y = 1

✗Common mistakes

Don’t write	Write instead
fg(2): working out f first	fg means f(g(2)): do g first
f⁻¹(x) = 1/f(x)	The inverse undoes f; it is not the reciprocal
gf(x) = 2x + 1²	Use brackets: (2x + 1)²
2f(x) and f(2x) treated as the same	2f(x) doubles the output; f(2x) doubles the input

★In the exam

Use brackets every time you substitute an expression.
Check an inverse with a number: f(2) then f⁻¹ of the answer should give 2.
For 'solve ff(x) = …' find ff(x) as an expression first.