Functions Worked Examples for IGCSE Maths
Working through solved examples is one of the most effective ways to master functions in IGCSE Mathematics. These worked examples, curated by Teacher Rig, cover the most common question types you will
Working through solved examples is one of the most effective ways to master functions in IGCSE Mathematics. These worked examples, curated by Teacher Rig, cover the most common question types you will encounter in the Cambridge IGCSE 0580 exam. Each solution shows every step of working with clear explanations of the reasoning behind each step.
Example 1: Finding a composite function
Question
f(x) = 3x + 1 and g(x) = x squared. Find fg(2).
Solution
- 1
Apply g first (inner function)
g(2) = 2 squared = 4
fg(2) means f(g(2)). Apply g first.
- 2
Apply f to the result
f(4) = 3(4) + 1 = 13
Now substitute g(2) = 4 into f.
Final Answer: fg(2) = 13
Exam Tip
For fg(x), always apply g first then f. Read from right to left: f of g of x.
Example 2: Finding the inverse function
Question
f(x) = (2x - 5) / 3. Find f inverse of x.
Solution
- 1
Write y = f(x)
y = (2x - 5) / 3
Replace f(x) with y.
- 2
Swap x and y
x = (2y - 5) / 3
This is the key step for finding inverses.
- 3
Solve for y
3x = 2y - 5, then 3x + 5 = 2y, then y = (3x + 5) / 2
Multiply both sides by 3, add 5, divide by 2.
Final Answer: f inverse of x = (3x + 5) / 2
Exam Tip
Check your inverse by verifying f(f inverse of x) = x. Here: f((3x+5)/2) = (2(3x+5)/2 - 5)/3 = (3x+5-5)/3 = 3x/3 = x. Correct!
Example 3: Finding the composite function expression
Question
f(x) = 2x + 3 and g(x) = x squared - 1. Find (a) fg(x), (b) gf(x).
Solution
- 1
Find fg(x)
fg(x) = f(g(x)) = f(x squared - 1) = 2(x squared - 1) + 3 = 2x squared - 2 + 3 = 2x squared + 1
Replace every x in f with the expression g(x) = x squared - 1.
- 2
Find gf(x)
gf(x) = g(f(x)) = g(2x + 3) = (2x + 3) squared - 1 = 4x squared + 12x + 9 - 1 = 4x squared + 12x + 8
Replace every x in g with the expression f(x) = 2x + 3.
Final Answer: fg(x) = 2x squared + 1, gf(x) = 4x squared + 12x + 8
Exam Tip
Notice fg(x) and gf(x) give different results. Order matters in composition!
Example 4: Domain and range from a graph
Question
f(x) = x squared - 4x + 3 for 0 <= x <= 5. Find the range of f.
Solution
- 1
Find the minimum point
x = -b/2a = 4/2 = 2. f(2) = 4 - 8 + 3 = -1
The minimum of a quadratic ax squared + bx + c occurs at x = -b/(2a).
- 2
Find the values at the endpoints
f(0) = 0 - 0 + 3 = 3. f(5) = 25 - 20 + 3 = 8.
Check the function value at both ends of the domain.
- 3
State the range
The minimum value is -1 (at x = 2) and the maximum is 8 (at x = 5). Range: -1 <= f(x) <= 8.
The range includes all output values from the minimum to the maximum.
Final Answer: Range: -1 <= f(x) <= 8
Exam Tip
For the range of a quadratic on a restricted domain, always check the vertex and both endpoints.
Explore Functions Subtopics
Frequently Asked Questions
How many functions questions appear in the IGCSE exam?
Functions typically appears in both Paper 2 (non-calculator) and Paper 4 (calculator). You can expect 2-4 questions on functions across both papers, worth a combined 15-25 marks depending on the session.
What is the best way to practise functions for IGCSE?
Start by understanding the methods through worked examples like these, then practise past paper questions under timed conditions. Teacher Rig recommends working through at least 20 functions past paper questions before your exam, checking your method against mark schemes.
Should I memorise functions formulas for the exam?
Some formulas are given on the formula sheet in the exam, but you should still be very familiar with them. Key formulas that are NOT on the sheet should be memorised. Practice using the formulas so that applying them becomes automatic.
Need Help with Functions?
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