IGCSE Maths / Functions and function notation
Excluded values from a domain revision notes
Edexcel International GCSE Maths (4MA1), chapter 24: Functions and function notation, specification point 3.2c. Free lesson.
Watch the video lesson · IGCSE Maths past papers
Some values cannot be used as inputs because they make a fraction divide by zero or put a negative number under a square root. Finding these excluded values helps you state the domain of a function correctly.
How a function can break
At this level, two things can make a function undefined. For a fraction, the denominator cannot be zero because division by zero has no answer. For a square root, the expression inside cannot be negative. These problems need different methods, so first identify which one you have. A fraction excludes individual values, written as x ≠ a value. A square root excludes a whole range of values, written as an inequality.
Why a zero denominator is excluded
For the reciprocal function y = 1/x, x = 0 is impossible because it would mean dividing 1 by 0. The graph has an asymptote, a line that the curve gets closer to but never touches, at x = 0. As x approaches zero from above, y grows without settling on a value. At x = 0.5, y = 2. At x = 0.1, y = 10. The graph also gets closer to y = 0 but never reaches it. The excluded input is x = 0, where the denominator is zero.
How to find excluded values in a fraction
Only the denominator matters. Set it equal to zero and solve. Check that your answer really makes the denominator zero, then write the value as excluded, not included.
How to find excluded values under a square root
The expression under a square root must be zero or greater. Set that expression ≥ 0 and solve the inequality. Values below the resulting boundary are excluded.
Check for two excluded values
When solving for a squared x, remember that both a positive and a negative value can give the same square. For example, x² = 9 has two solutions, so both make the denominator zero and must be excluded. For 1/(x² − 9), x² − 9 = 0 gives x² = 9, so x = 3 or x = −3. The excluded values are written x ≠ 3 and x ≠ −3, or x ≠ ±3.