The domain of a function is the set of inputs it can take, and the range is the set of outputs it produces. Square roots need a non-negative input, logarithms need a positive input, and fractions cannot have a zero denominator. The range comes from the shape of the function: completing the square for quadratics, the behaviour at each end for reciprocals, and the limits of −1 and 1 for sine and cosine.
Domain rules and useful facts
- √g(x) needs g(x) ≥ 0
- ln g(x) needs g(x) > 0
- 1/g(x) needs g(x) ≠ 0
- eˣ > 0 for all x
- sin x and cos x lie between −1 and 1
(x + p)² + q has least value q, so its range is f(x) ≥ q.
What are the domain and range?
The domain is the set of x-values a function is allowed to take, read along the x-axis of its graph. The range is the set of outputs it actually produces, read up the y-axis. Use f(x) or y in the range, for example f(x) ≥ 2, not x ≥ 2.
When finding a range over a given domain, check the endpoints and any turning point inside the domain. The range runs from the lowest output to the highest, which is not always at the ends.
How do you find the domain of a function?
Some expressions only accept certain inputs. A square root needs whatever is inside to be at least zero, a logarithm needs its input to be strictly positive, and a fraction cannot have a zero denominator. Write the condition as an inequality or an exclusion and solve it for x: the solution is the domain.
Range of quadratics and root functions
For a quadratic, complete the square. Since (x + p)² is never negative, (x + p)² + q has least value q, so the range is f(x) ≥ q for a U-shaped curve. For example, x² + 4x + 9 = (x + 2)² + 5, so its range is f(x) ≥ 5. Over a restricted domain, compare the turning point with the values at the endpoints.
A square root is never negative, so √(something) ≥ 0. Subtracting a root, as in a − √x, reverses things: the largest output comes where the root is smallest.
Range of reciprocal functions
For f(x) = 1/g(x), find the range of the denominator g(x) first, then invert it. Where the denominator is large, the reciprocal is close to 0; where it is close to 0, the reciprocal becomes very large. A reciprocal can never equal 0, so ranges like 0 < f(x) ≤ 1 often appear with a strict inequality at one end.
Reciprocals involving eˣ and ln x
Two facts drive these questions: eˣ is always positive, tending to 0 as x → −∞ and growing without limit as x → ∞; and ln x is only defined for x > 0, tending to −∞ as x approaches 0. Work out how the denominator behaves at both ends of the domain. Those two limits bound the reciprocal, and limits that are approached but not reached give strict inequalities.
Range of trig functions
Over a full period, sin x and cos x run between −1 and 1, so a sin x + b runs between b − a and b + a (for positive a). On a restricted domain, the trig part may not reach both −1 and 1, so find its actual greatest and least values over that interval first.
Worked examples
Example 1. The function f(x) = 5 − (x − 1)² has domain −1 ≤ x ≤ 3. Write down the domain and range of f.
- The domain is the given interval: −1 ≤ x ≤ 3.
- The maximum is at the vertex (1, 5).
- The ends give f(−1) = f(3) = 1, the least values.
Answer: Domain −1 ≤ x ≤ 3; range 1 ≤ f(x) ≤ 5.
Marks: B1 for the domain, B1 for the range.
Example 2. Find the domain of f(x) = √(x − 3).
- The inside of the square root must be at least zero: x − 3 ≥ 0.
- So x ≥ 3.
Answer: x ≥ 3.
Marks: M1 for x − 3 ≥ 0, A1 for x ≥ 3.
Example 3. The function f(x) = 1/(x² + 1) has domain all real x. Find its range.
- The denominator x² + 1 ≥ 1, with least value 1 at x = 0.
- The largest output is f(0) = 1.
- As x → ±∞ the denominator grows without limit, so f → 0, but never reaches it.
Answer: 0 < f(x) ≤ 1.
Marks: M1 for the range of the denominator, A1 for inverting it.
Example 4. The function f(x) = 1/(eˣ + 1) has domain all real x. Find its range.
- eˣ > 0, so eˣ + 1 > 1, and it can be as large as you like.
- As x → −∞, f → 1, not reached. As x → ∞, f → 0, not reached.
- So f takes every value strictly between 0 and 1.
Answer: 0 < f(x) < 1.
Marks: M1 for bounding the denominator, A1 for inverting.
Example 5. The function f(x) = 3 sin x + 1 has domain 0 ≤ x ≤ 2π. Find its range.
- Over a full period, sin x runs from −1 to 1.
- Least value: 3(−1) + 1 = −2. Greatest: 3(1) + 1 = 4.
Answer: −2 ≤ f(x) ≤ 4.
Marks: M1 for −1 ≤ sin x ≤ 1, A1 for the range.
Common mistakes
- Writing the range in terms of x. The range describes outputs: write f(x) ≥ 2 or y ≥ 2, not x ≥ 2.
- Using only the endpoints for the range. A turning point inside the domain can give the maximum or minimum.
- Including zero in the range of a reciprocal. 1/g(x) can never equal 0, so the inequality at 0 must be strict.
- Using ≥ instead of > for a logarithm. ln g(x) needs g(x) > 0; zero is not allowed.
- Assuming sin x reaches ±1 on a short interval. On a restricted domain, find the actual extreme values of the trig part.
How domain and range are examined
On the Edexcel International A-Level Pure 3 paper (WMA13), domain and range questions belong to the functions and graphs chapter. In Edexcel A-Level (9MA0) they are part of second-year Pure. You will be asked to:
- state the range of a function, often one involving eˣ, ln x or a completed square
- find the largest possible domain of a function
- use the range of f to give the domain of its inverse
- explain why a function is or is not one-to-one on a given domain
The Pure 3 functions and graphs exam questions cover these with full mark schemes. For more practice there is a functions and graphs worksheet, and the algebra revision engine creates new questions each time.
The range of a function becomes the domain of its inverse, which is the focus of composite and inverse functions.
Frequently asked questions
What is the difference between domain and range?
The domain is the set of inputs, x-values. The range is the set of outputs, f(x)-values.
How do you find the range of a quadratic?
Complete the square. (x + p)² + q has minimum value q, so the range is f(x) ≥ q, unless the domain is restricted.
Why can't the range of 1/g(x) include 0?
A fraction with numerator 1 can never equal 0, however large the denominator becomes. It only approaches 0.