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Master f(x), fg(x) and f inverse for IGCSE Higher. The golden rule, swap-and-solve method and worked examples for grades 7-9.
A composite function fg(x) means applying g first, then f — work from right to left. An inverse function f inverse reverses the original: if f maps input to output, f inverse maps output back to input. Use the Swap-and-Solve method to find inverses: write y = f(x), swap x and y, rearrange for y, then rename as f inverse of x. Both topics are Higher-tier only.
Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1
A function is a mathematical machine: it takes an input, applies a rule, and produces an output. Function notation uses f(x) to name the rule, where x is the input variable. For example, f(x) = 2x + 3 means "double the input and add three".
You can use any letter for the function name — f, g and h are the most common in IGCSE questions. To evaluate a function at a specific value, replace every x with that value: f(4) = 2(4) + 3 = 11.
The domain is the set of all valid inputs for a function. The range is the set of all possible outputs. For example, if f(x) = 1/x, the domain excludes x = 0 because division by zero is undefined. The range also excludes 0 because 1/x can never equal zero.
At IGCSE level, domain restrictions most commonly arise from denominators (cannot equal zero) and square roots (expression inside must be non-negative). Always check whether the question specifies a restricted domain such as x > 0.
A composite function chains two functions together. The notation fg(x) means "apply g first, then apply f to the result". The golden rule is:
Work from right to left: fg(x) = f(g(x))
To find fg(x) as a formula, substitute the entire expression for g(x) wherever you see x in f(x). To find fg at a specific value, first calculate g of that value, then feed the answer into f.
Imagine f converts dirhams to dollars and c adds a bank fee. The order matters: fc(x) applies the fee first then converts, while cf(x) converts first then adds the fee. These give different results — just as fg(x) and gf(x) almost never produce the same answer.
The inverse function f⁻¹(x) reverses the original function. If f maps a to b, then f⁻¹ maps b back to a. Use the Swap-and-Solve method:
This method works for all functions at IGCSE level, including those with fractions. The key algebraic challenge is the rearranging step, which may require cross-multiplication and collecting terms.
f(x) = x² + 1, g(x) = 2x - 3. Find fg(4).
Step 1 (Apply g first): g(4) = 2(4) - 3 = 8 - 3 = 5
Step 2 (Feed into f): f(5) = 5² + 1 = 25 + 1 = 26
Answer: fg(4) = 26
f(x) = 3x + 2, g(x) = x². Find gf(x) as a simplified expression.
Step 1 (Identify inner function): gf(x) means apply f first, then g
Step 2 (Find f(x)): f(x) = 3x + 2
Step 3 (Substitute into g): gf(x) = g(3x + 2) = (3x + 2)²
Step 4 (Expand): (3x + 2)² = 9x² + 12x + 4
Check: gf(1) = (3 + 2)² = 25. Formula: 9 + 12 + 4 = 25. Correct.
Answer: gf(x) = 9x² + 12x + 4
Find f⁻¹(x) for f(x) = (2x + 1) / (x - 3).
Step 1 (Write as y =): y = (2x + 1) / (x - 3)
Step 2 (Swap x and y): x = (2y + 1) / (y - 3)
Step 3 (Cross-multiply): x(y - 3) = 2y + 1
Step 4 (Expand): xy - 3x = 2y + 1
Step 5 (Collect y terms): xy - 2y = 3x + 1
Step 6 (Factor out y): y(x - 2) = 3x + 1
Step 7 (Divide): y = (3x + 1) / (x - 2)
Answer: f⁻¹(x) = (3x + 1) / (x - 2)
Order confusion (fg vs gf): fg(x) means apply g first, then f. gf(x) means apply f first, then g. These are almost never the same. Always read right to left.
Inverse notation confusion: f⁻¹(x) does NOT mean 1/f(x). The superscript -1 denotes the inverse function, not a reciprocal. The inverse undoes the original function; the reciprocal divides 1 by it.
Domain errors with denominators: When finding inverses of fractional functions, the new denominator creates a domain restriction. If f⁻¹(x) = (3x + 1)/(x - 2), then x = 2 is excluded from the domain of the inverse.
Forgetting to swap before rearranging: In the Swap-and-Solve method, if you rearrange first without swapping x and y, you will get x in terms of y — which is the original function rearranged, not the inverse.
Use substitution to verify composites: After finding fg(x) as a formula, check by evaluating at a simple value like x = 1. Calculate g(1) then f of that result — it should match your formula at x = 1.
Check inverses with ff⁻¹(x) = x: If your inverse is correct, then f(f⁻¹(x)) must simplify to x. This is a powerful exam check that catches algebraic slips.
Label every step clearly: Examiners look for method marks. Write "swap x and y", "cross-multiply", "collect y terms" as annotations so the examiner can follow your logic even if you make an arithmetic error.
Read the question carefully: "Find fg(x)" asks for a general formula. "Find fg(3)" asks for a single number. The approach differs — formula questions require algebraic substitution, while numerical questions are calculated step by step.
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fg(x) means the composite function "f of g of x". You apply g first, then feed the output into f. The golden rule is to work from right to left. For example, if f(x) = x squared + 1 and g(x) = 2x - 3, then fg(4) means first calculate g(4) = 5, then f(5) = 26.
Use the Swap-and-Solve method: write the function as y = f(x), swap x and y so x becomes the subject, rearrange to isolate y, then rename y as f inverse of x. For example, if f(x) = 3x + 2, write y = 3x + 2, swap to get x = 3y + 2, solve for y = (x - 2)/3, so f inverse of x = (x - 2)/3.
No. Function notation, composite functions and inverse functions are Higher-tier only topics in Edexcel IGCSE Mathematics (4MA1). They do not appear on the Standard (Foundation) tier papers. However, understanding function notation is essential preparation for A-Level Mathematics.
A specialist IGCSE maths tutor identifies whether the difficulty lies in understanding function notation, applying the correct order for composites, or handling the algebra of inverses with fractions. GetYourTutors provides full-time, in-home tutors across Dubai who build confidence with functions from Grade 7 through Grade 9.
A dedicated tutor builds confident composite and inverse function skills step by step. In-home across Dubai.