Functions: Domain and Range Restrictions
In IGCSE Algebra, functions have a domain (the set of inputs a function is allowed to take) and a range (the set of outputs the function produces). Some restrictions are forced by the algebra: a denominator cannot be zero, and the square root of a negative number is not real, so those inputs are excluded from the domain. Other restrictions are imposed by the question, and then the range must be worked out for that restricted domain rather than for all real numbers. Finding a range is a matter of asking what the function can actually produce: a squared term is never negative, a square root is never negative, and a quadratic has a least or greatest value at its turning point.
Before you start
Make sure you're comfortable with these topics first:
Method
- Look for the two forced restrictions first: set any denominator equal to zero and exclude those x values, and require the expression under any square root to be greater than or equal to zero.
- Write the domain using inequalities or set notation, saying x is not equal to a value where a denominator vanishes, or x is greater than or equal to a value where a root begins.
- For the range, identify the shape of the function. A squared term has a minimum of zero, so a quadratic in completed square form shows its least or greatest value immediately.
- For a restricted domain, evaluate the function at the ENDPOINTS of the domain and check whether there is a turning point inside it, because the extreme values of the range occur at one or the other.
- For a function with a denominator, note which value the output can never take: y = 1/(x - 2) can produce any value except zero, and adding a constant shifts that excluded value.
- State the range in the same style as the domain, using inequalities in terms of f(x) or y, and check plausibility by testing one or two inputs.
Worked example
f(x) = the square root of (x - 4). (a) State the domain of f. (b) State the range of f. (c) For g(x) = x squared - 6x + 11 with domain all real numbers, find the range of g.
- For the domain of f, the expression under the root must not be negative: x - 4 is greater than or equal to 0, so x is greater than or equal to 4.
- For the range of f, a square root is never negative and can be made as large as wanted by taking a large x, so f(x) is greater than or equal to 0.
- For g, complete the square: x squared - 6x + 11 = (x - 3) squared - 9 + 11 = (x - 3) squared + 2.
- The squared bracket is never negative, and it is zero when x = 3.
- So the least value of g is 0 + 2 = 2, occurring at x = 3, and g increases without limit either side.
- Final answers: the domain of f is x greater than or equal to 4, the range of f is f(x) greater than or equal to 0, and the range of g is g(x) greater than or equal to 2.
Practice questions
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Q1State the domain of f(x) = 1/(x + 5).Show answer
Answer: All real x except x = -5, because the denominator would be zero there.
Q2State the domain of f(x) = the square root of (x + 2).Show answer
Answer: x is greater than or equal to -2.
Q3State the range of f(x) = x squared.Show answer
Answer: f(x) is greater than or equal to 0.
Q4Write x squared + 4x + 7 in completed square form and state its least value.Show answer
Answer: (x + 2) squared + 3, so the least value is 3.
Q5State the range of f(x) = 1/x for x greater than 0.Show answer
Answer: f(x) is greater than 0, since a positive input gives a positive output that can be made as large or as close to zero as required but never reaches zero.
Q6f(x) = 2x + 1 with domain 0 is less than or equal to x is less than or equal to 5. Find the range.Show answer
Answer: The function is increasing, so evaluate the endpoints: f(0) = 1 and f(5) = 11, giving 1 is less than or equal to f(x) is less than or equal to 11.
Q7Why must the endpoints of a restricted domain be checked when finding a range?Show answer
Answer: For a function without a turning point inside the domain, the extreme output values occur at the ends of the domain, so those are the values that bound the range.
Exam-style questions
Written in the style of a IGCSE Maths exam paper, with a full mark scheme.
f(x) = x squared - 4x + 9 with domain 0 is less than or equal to x is less than or equal to 5. Find the range of f, showing your method.
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f(x) = 3/(x - 1) + 2. (a) State the value of x that must be excluded from the domain and explain why. (b) State the value that f(x) can never take, and explain why.
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Free printable worksheet
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