A function does not automatically accept every imaginable input, and it does not automatically produce every imaginable output. The domain describes the permitted inputs. The range describes the outputs the function actually produces.
This guide develops domain and range through formulas, graphs and real-world models, then shows why restrictions are not technical decorations. They are part of the mathematical definition of the function.
One-sentence answer
Domain tells us what may go in; range tells us what can come out.
A simple unrestricted-looking function
For f(x)=2x+1 over the real numbers, every real input is permitted and every real output is possible.
So its domain is all real numbers and its range is all real numbers.
A denominator creates a restriction
Consider:
f(x)=1/(x-3)The input x=3 is not permitted because it makes the denominator zero. Therefore the domain excludes 3.
The function can never output 0 either, because 1 divided by a finite non-zero number cannot equal 0. So the range also excludes 0.
A square-root restriction
Over the real numbers, consider:
g(x)=√(x-4)The expression inside the square root must be non-negative:
x-4 ≥ 0
x ≥ 4The outputs of a principal square root are also non-negative, so the range is y≥0.
Quadratic range
For h(x)=x², every real x is allowed. But x² can never be negative.
Therefore the domain is all real numbers and the range is y≥0.
Graphical meaning
On a graph, the domain is the set of horizontal x-values for which the graph exists. The range is the set of vertical y-values reached by the graph.
A graph can make restrictions visible immediately. A vertical asymptote may reveal an excluded input. A lowest point can reveal a lower bound on the range.
Real-world restrictions
Suppose C(n)=12n+50 models the cost of printing n posters. Algebraically, the formula works for any real n. In context, n may need to be a whole number greater than or equal to zero.
The meaningful domain therefore comes from the situation, not merely from the algebraic expression.
Why restrictions matter for inverse functions
The function f(x)=x² is many-to-one over all real numbers because f(3)=9 and f(-3)=9. That prevents a unique inverse function over the full real domain.
If we restrict the domain to x≥0, each output has one corresponding input and the inverse becomes f⁻¹(x)=√x.
This is one of the clearest reasons domain restrictions are mathematically important.
Worked example 1
Find the domain of:
f(x)=5/(2x+6)The denominator cannot equal zero:
2x+6 ≠ 0
x ≠ -3So the domain is all real x except -3.
Worked example 2
Find the domain of g(x)=√(10-2x).
10-2x ≥ 0
-2x ≥ -10
x ≤ 5The domain is x≤5.
Worked example 3: contextual domain
A tank contains V(t)=500-20t litres after t minutes of draining. The model is meaningful from the start until the tank reaches zero.
500-20t ≥ 0
t ≤ 25With time starting at t=0, the contextual domain is 0≤t≤25.
Diagnostic table
| Observed mistake | Likely issue | Repair |
|---|---|---|
| Says domain is always all real numbers | Restrictions ignored | Check denominators, roots and context |
| Confuses domain with range | Input-output roles | Ask “what can go in?” then “what can come out?” |
| Allows negative poster counts | Context detached from formula | Apply real-world constraints |
| Creates inverse of x² without restriction | Many-to-one mapping | Restrict domain before inverting |
A reliable routine
- Inspect the formula for algebraic restrictions.
- Inspect the context for physical or counting restrictions.
- State the permitted inputs clearly.
- Determine which outputs can actually occur.
- Check whether any proposed inverse is one-to-one on the chosen domain.
Practice
- Find the domain of f(x)=1/(x+4).
- Find the domain of g(x)=√(x+7).
- Find the range of h(x)=x²+5 over all real x.
- Explain why x² needs a restricted domain before it has a unique inverse function.
Answers
1. x≠-4. 2. x≥-7. 3. y≥5. 4. Because positive and negative inputs can give the same output, so reversing would not give one unique input unless the domain is restricted.
Connected routes
Use What a Function Really Is for the core definition and continue to Composite and Inverse Thinking Before Additional Mathematics. Return to the Mathematics Learning Hub.