Functions: Domain and Range Explained Simply

What Are Domain and Range?

Every function takes an input and produces an output. The domain is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values). At IGCSE Extended level, you need to be comfortable identifying both from an equation, a graph, or a description.

Think of a function as a machine. The domain is everything you are allowed to feed into the machine. The range is everything the machine can produce. Some inputs might break the machine — those values are excluded from the domain.

Notation You Need to Know

Cambridge uses a specific notation for domain and range. You will usually see something like:

f(x) = 2x + 1, for x > 0 — here the domain is explicitly stated as all values greater than 0
The range of f is f(x) ≥ 5 — this tells you the smallest output value is 5

You may also see set notation or inequality notation. Both are acceptable unless the question specifies a particular form.

Finding the Domain

Linear Functions

For a simple linear function like f(x) = 3x − 2, the domain is all real numbers unless the question restricts it. There is nothing you can substitute for x that would cause a problem — no division by zero, no square root of a negative number.

Functions with Fractions

When a function has x in the denominator, you must exclude values that make the denominator zero.

Worked Example 1: Find the domain of f(x) = 5 / (x − 3).

The denominator is x − 3. Setting x − 3 = 0 gives x = 3. At x = 3, the function is undefined because we cannot divide by zero.

Domain: x ∈ ℝ, x ≠ 3

Functions with Square Roots

The expression inside a square root must be greater than or equal to zero (assuming we are working with real numbers only, which is always the case at IGCSE).

Worked Example 2: Find the domain of f(x) = √(2x − 6).

Set the expression inside the root ≥ 0:

2x − 6 ≥ 0

2x ≥ 6

x ≥ 3

Domain: x ≥ 3

Combined Restrictions

Sometimes a function has both a fraction and a root, or the question gives additional restrictions. Apply all constraints together.

Worked Example 3: f(x) = 1 / √(x − 1), find the domain.

The square root requires x − 1 ≥ 0, so x ≥ 1. But the square root is also in the denominator, so √(x − 1) ≠ 0, meaning x ≠ 1. Combining these: x > 1.

Finding the Range

From a Graph

The easiest way to find the range is from a graph. Look at the y-axis and identify the lowest and highest points the curve reaches. If the graph extends upward or downward forever, the range is unbounded in that direction.

From a Quadratic Function

Quadratic functions have a turning point, which gives either a minimum or maximum y-value.

Worked Example 4: Find the range of f(x) = x² − 4x + 7.

Complete the square: x² − 4x + 7 = (x − 2)² − 4 + 7 = (x − 2)² + 3.

Since (x − 2)² ≥ 0 for all x, the smallest value of f(x) is 0 + 3 = 3.

Range: f(x) ≥ 3

Worked Example 5: Find the range of f(x) = −2x² + 8x − 5.

Factor out −2 from the first two terms: −2(x² − 4x) − 5

Complete the square inside: −2[(x − 2)² − 4] − 5 = −2(x − 2)² + 8 − 5 = −2(x − 2)² + 3

Since −2(x − 2)² ≤ 0, the maximum value of f(x) is 3.

Range: f(x) ≤ 3

From Other Function Types

For exponential functions like f(x) = 2ˣ, the output is always positive, so the range is f(x) > 0. If the function is f(x) = 2ˣ + 3, the range shifts up to f(x) > 3.

For trigonometric functions, sin(x) and cos(x) always produce values between −1 and 1 inclusive, so the range of f(x) = sin(x) is −1 ≤ f(x) ≤ 1.

Domain and Range with Restricted Domains

A common exam question gives a function with a restricted domain and asks for the range.

Worked Example 6: f(x) = x² − 2x for −1 ≤ x ≤ 4. Find the range of f.

Step 1: Complete the square. f(x) = (x − 1)² − 1. The turning point is at (1, −1).

Step 2: Since x = 1 lies within the domain [−1, 4], the minimum value is −1.

Step 3: Check the endpoints.

f(−1) = (−1)² − 2(−1) = 1 + 2 = 3
f(4) = (4)² − 2(4) = 16 − 8 = 8

Step 4: The maximum output is 8 and the minimum is −1.

Range: −1 ≤ f(x) ≤ 8

Common Mistakes to Avoid

Forgetting to check both endpoints and the turning point when the domain is restricted. The maximum or minimum might occur at an endpoint, not the vertex.
Including values where the function is undefined. If the denominator equals zero at a certain x, that x-value is not in the domain.
Confusing domain and range. Domain is always about x (inputs). Range is always about y or f(x) (outputs).
Writing strict inequalities when they should be inclusive, or vice versa. Pay attention to whether the boundary value is actually achieved.

Exam Tips

If a graph is provided, use it. Read the range directly from the y-axis.
For quadratics, always complete the square to find the turning point — this gives you the boundary of the range.
When asked for domain, look for division by zero and square roots of negative numbers first.
Write your final answer clearly using correct inequality notation.

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