What is a function?
A function is a rule that assigns to each input value exactly one output value. We often write f(x), read as “f of x”. The set of allowed inputs is the domain, and the set of resulting outputs is the range.
Key idea
For y = f(x): the domain is the set of x-values you may substitute; the range is the set of y-values produced. A relation is a function only if every x gives one and only one y.
Restrictions on the domain
Some expressions are undefined for certain inputs. You cannot divide by zero, so 1⁄(x−2) excludes x = 2. You cannot take the square root of a negative number in real work, so √(x−4) needs x ≥ 4. Watching for these gives the natural domain.
Finding a range
For a linear function the range is found from the end values of the domain. For a quadratic, the vertex gives the least or greatest value.
Worked example
Let f(x) = 2x + 3 with domain 0 ≤ x ≤ 4. Since f is increasing, the smallest output is f(0) = 3 and the largest is f(4) = 11. Therefore the range is 3 ≤ f(x) ≤ 11.
Remember
- One input → exactly one output.
- Exclude values that make a denominator zero or a square root negative.
- For x² type functions, the range has a minimum (or maximum) at the vertex.