Connect an input rule, its stated domain and its range.
Notes and key methods
A function assigns exactly one output to every input in its domain. Different inputs may share an output; one input cannot have two outputs. The range consists of outputs actually attained on the stated domain.
For a non-constant linear rule \(f(x)=ax+b\), endpoint images determine the range on a closed interval. Reverse their order if a is negative. A one-to-one rule has an inverse after domain and range are interchanged.
Check the conditions
Use the domain given in the question.
A range endpoint is included only when it is attained.
Three worked examples
Example 1 · Functions, domains and ranges
\(f{\left(x \right)} = 4 x + 5\), with domain \(\left[3, 9\right]\). Write \(\displaystyle a=4\) for its gradient and u for its lower domain endpoint.
State the range of f, including its endpoints.
Explain why f is one-to-one on its domain and find its inverse rule.
Find the input whose output is f(u)+a.
Find the length of the range interval.
Show the reasoning
(a)
The gradient 4 is positive, so the endpoint images are 17 and 41.
Answer: \(\left[17, 41\right]\)
(b)
The nonzero gradient makes f strictly increasing. Solve \(\displaystyle y=4x+5\) for x, then interchange input and output; the inverse domain is the range just found.
Subtract the least output from the greatest output.
\(24\)
Answer: \(24\).
Example 2 · Functions, domains and ranges
\(f{\left(x \right)} = 5 x + 6\), with domain \(\left[1, 8\right]\). Write \(\displaystyle a=5\) for its gradient and u for its lower domain endpoint.
State the range of f, including its endpoints.
Explain why f is one-to-one on its domain and find its inverse rule.
Find the input whose output is f(u)+a.
Find the length of the range interval.
Show the reasoning
(a)
The gradient 5 is positive, so the endpoint images are 11 and 46.
Answer: \(\left[11, 46\right]\)
(b)
The nonzero gradient makes f strictly increasing. Solve \(\displaystyle y=5x+6\) for x, then interchange input and output; the inverse domain is the range just found.
Subtract the least output from the greatest output.
\(35\)
Answer: \(35\).
Example 3 · Functions, domains and ranges
\(f{\left(x \right)} = 2 x + 7\), with domain \(\left[2, 6\right]\). Write \(\displaystyle a=2\) for its gradient and u for its lower domain endpoint.
State the range of f, including its endpoints.
Explain why f is one-to-one on its domain and find its inverse rule.
Find the input whose output is f(u)+a.
Find the length of the range interval.
Show the reasoning
(a)
The gradient 2 is positive, so the endpoint images are 11 and 19.
Answer: \(\left[11, 19\right]\)
(b)
The nonzero gradient makes f strictly increasing. Solve \(\displaystyle y=2x+7\) for x, then interchange input and output; the inverse domain is the range just found.