Function Notation
Edexcel International GCSE Mathematics A· 3.2· 12 min read
1. Function Definition and Core Notation★★☆☆☆Higher only⏱ 3 min
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Function
A rule that maps each input value from one set to exactly one output value in a second set.
Example:
f(x) = 2x + 3 maps each input x to an output equal to twice x plus 3.
You will encounter two standard function notations in your exam: the common f(x) format, and the mapping format f : x ↦ ... Both describe the same function rule, and you may be asked to convert between them or use either for calculations.
The function f is defined as f : x ↦ 5x - 7. Write this function using f(x) notation, and calculate f(4).
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Step 1: Convert mapping notation to f(x) notation by replacing the ↦ symbol with an equals sign after f(x)
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Step 2: Substitute x = 4 into the function rule to calculate f(4)
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Exam tip:
Always confirm the notation requested in the question before submitting your answer, as marks are awarded for correct use of either f(x) or mapping format as specified.
2. Domain and Range of Functions★★★☆☆Higher only⏱ 3 min
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Domain and Range
Domain = the set of all valid input values (x) for a function. Range = the set of all resulting output values (f(x)) for the function.
Example:
For f(x) = 1/(x - 2), x = 2 is excluded from the domain because dividing by zero is undefined.
The most common reason to exclude values from a domain is when they make a denominator equal to zero, as division by zero is mathematically undefined. You may also see exclusions for values that produce a square root of a negative number, though these are less common in this sub-topic for Edexcel IGCSE.
State the value excluded from the domain of the function g(x) = 4/(3x + 9), and describe the range of h(x) = x² + 2.
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Step 1: Find the value of x that makes the denominator of g(x) equal to zero
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Step 2: Excluded domain value for g(x) is x = -3
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Step 3: For h(x) = x² + 2, x² is always ≥ 0 for all real x, so x² + 2 ≥ 2
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Step 4: Range of h(x) is all real numbers greater than or equal to 2, written as h(x) ≥ 2
3. Composite Functions★★★★☆Higher only⏱ 3 min
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Composite Function fg
The function formed by applying function g first, then substituting the result into function f. fg(x) = f(g(x)).
Example:
If f(x) = x + 1 and g(x) = 2x, then fg(x) = f(2x) = 2x + 1.
The order of composite functions is critical for Edexcel IGCSE: fg is always g first, then f, which is the opposite of the order the letters are written. Failing to follow this order is one of the most common mistakes on exam questions for this topic.
Given f(x) = 3x + 2 and g(x) = x² - 1, calculate fg(2) and write an expression for fg(x) in its simplest form.
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Step 1: First calculate g(2) to get the input for f
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Step 2: Substitute g(2) = 3 into f(x)
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Step 3: To find the general fg(x), substitute g(x) into f(x) wherever x appears
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4. Inverse Functions★★★★☆Higher only⏱ 3 min
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Inverse Function f⁻¹(x)
The function that reverses the action of the original function f. If f(x) maps input x to output y, then f⁻¹(y) maps y back to x.
Example:
If f(x) = 2x, then f⁻¹(x) = x/2, as dividing by 2 reverses the action of multiplying by 2.
The Edexcel specified method to find an inverse function is fixed: write y = f(x), swap x and y, then rearrange to make y the subject of the equation. This resulting y is your inverse function f⁻¹(x).
Find the inverse function of f(x) = (4x + 1)/3.
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Step 1: Write the function as y = f(x)
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Step 2: Swap x and y in the equation
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Step 3: Rearrange to make y the subject
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Step 4: Replace y with f⁻¹(x)
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5. Common Pitfalls
Wrong move:
Applying f first then g when calculating fg(x)
Why:
The Edexcel spec defines fg(x) as f(g(x)), so g is the first operation applied, not f.
Correct move:
Always substitute the inner function (the one written second) into the outer function first.
Wrong move:
Forgetting to exclude x values that make denominators zero from the domain
Why:
Division by zero is undefined, so these values are not valid inputs for the function.
Correct move:
Set any denominator equal to zero and solve for x to find excluded domain values.
Wrong move:
Rearranging first then swapping x and y when calculating inverse functions
Why:
This will produce an incorrect inverse that does not reverse the action of the original function.
Correct move:
Follow the spec method strictly: write y = f(x), swap x and y, then rearrange for y.
Wrong move:
Confusing f⁻¹(x) with 1/f(x)
Why:
The superscript -1 in inverse function notation is not an exponent, it indicates the reverse function.
Correct move:
Recognize that f⁻¹(x) always refers to the inverse function, not the reciprocal of f(x).
Wrong move:
Writing the range as x ≥ k instead of f(x) ≥ k or y ≥ k
Why:
The range describes output values of the function, not input x values.
Correct move:
Always label range values with the function name or y, e.g. f(x) ≥ 2.
6. Quick Reference Cheatsheet
Concept | Rule/Notation | Exam Reminder |
|---|---|---|
Function notation | f(x) = ..., f:x ↦ ... | Both notations are equivalent, use whichever is requested |
Domain | Set of valid input x values | Exclude x that make denominators zero |
Range | Set of output f(x) values | Label range with f(x) or y, not x |
Composite function fg | fg(x) = f(g(x)) | Apply g first, then f; order matters |
Inverse function f⁻¹(x) |
| f⁻¹(x) ≠ 1/f(x); check by testing f(f⁻¹(x)) = x |
7. Frequently Asked
What order do I apply composite functions fg?
fg(x) means apply g(x) first, then substitute the result into f(x). This is the required convention for Edexcel IGCSE Maths, and marks are deducted for reversing the order.
How do I find the inverse of a function for my exam?
Follow the 3 spec-mandated steps: 1. Write y = f(x) 2. Swap x and y in the equation 3. Rearrange to make y the subject; this is f⁻¹(x).
What values do I need to exclude from a function's domain?
Exclude any x values that make a denominator zero (division by zero is undefined) or that produce a square root of a negative number, though square root cases are rare for this sub-topic.
Is f⁻¹(x) the same as 1/f(x)?
No. The superscript -1 in inverse function notation is not an exponent. It refers only to the function that reverses the action of f(x), not the reciprocal of f(x).
What's Next
Now you have mastered function notation for Edexcel IGCSE Higher Maths, you can move on to related higher-tier topics that build on this foundation. Next, you will learn to apply function rules to transform graphs of linear, quadratic, and trigonometric functions, a key part of the Sequences, Functions and Graphs unit. You will also use function notation in later topics including algebraic manipulation and calculus foundations for your higher-tier exam. Practicing mixed function questions that combine composite, inverse, and domain skills will help you prepare for the longer 4-6 mark questions that often appear on Paper 2H and 4H. Make sure you can apply all rules without a calculator as well, as some questions may appear on non-calculator papers even if the sub-topic allows calculators.