Domain and Range of a Function
Simple Explanation
The domain of a function is the complete set of allowed input values (x-values); the range is the complete set of output values (y-values) the function actually produces.
Why Do We Need It?
Knowing a function's domain tells you which inputs are valid to use — plugging in a value outside the domain (like dividing by zero, or taking the square root of a negative number) produces an undefined result.
Worked Example
Find the domain of a function with a restriction
Find the domain of f(x) = 1 / (x − 3).
Why Does This Work?
The domain must exclude exactly the inputs that break the function's definition (like division by zero) — checking for those specific trouble spots (zero denominators, negative numbers under an even root) reliably finds every value that must be excluded.
Real-Life Example
A parking fee function
A parking garage charges by the hour, and its fee function is only defined for a non-negative number of hours parked.
The domain of that fee function is naturally restricted to hours ≥ 0 — negative time parked does not make sense, just as it would not be part of the function's domain.
Practice
What must be excluded from the domain of g(x) = 1/(x + 5)?
MediumCommon mistake
Forgetting to check for domain restrictions on functions involving fractions or roots — not every function has "all real numbers" as its domain.
Quick Review
- Domain = the set of all valid inputs; range = the set of all resulting outputs.
- Exclude any input that causes division by zero or an even root of a negative number.
- Always check for these restrictions before assuming the domain is all real numbers.