A function gives each input one output
Function notation names a rule: it takes an input and produces an output. To evaluate, substitute the input for , e.g. for , .
Domain = the inputs; range = the outputs
The domain is the set of allowed inputs (-values) and the range is the set of resulting outputs (-values). Feed every domain value through the rule and collect the outputs to get the range.
A quadratic's range turns at the vertex
For the outputs are never negative, so over a domain like the range runs from the minimum (at ) up to the largest end value . The smallest output of a function is at its vertex, not at an endpoint.
Drawn from real examiner reports.
Confusing domain with range
Students swap the two: the domain is the -inputs you are allowed to use; the range is the -outputs you get back. Reading the wrong one is the most frequent slip on these questions.
Taking a curve's range from endpoints only
For a non-linear function the largest or smallest output may occur inside the domain, not at an end. For on the minimum is at (giving ), which an endpoints-only check would miss.
Reading f(4) as f × 4
does not mean — it means substitute for in the rule. For , , not .
Giving the domain back as the range
The range is the set of outputs, not the inputs. For on domain the range is — feeding each input through the rule — not the domain repeated back.
Squaring a negative input wrongly
For , square first: . Do not write or treat as — square the input before multiplying.
Linear range comes from the endpoints
A straight-line function is increasing or decreasing throughout, so over the range is just between and — evaluate both ends and the range spans them.
For a curve, check the turning point
For a non-linear function the largest or smallest output can be at a turning point inside the domain, not at an end. For on the minimum is at , which an endpoints-only check misses.
Feed every domain value through the rule
To find a range, put each allowed input through the function and collect the outputs. For a small listed domain, evaluate every value; for an interval, use the endpoints (and any turning point in between).
A function is a rule that turns each input into exactly one output. We write it as ("f of x"). To evaluate, substitute the input for : Other examples from the syllabus: and .
For f(x) = 3x − 5, what is f(4)?
A function is defined by . Work out .
