Functions, domains, ranges and linear inverses — 0580
Exchange the permitted inputs and attained outputs when reversing a one-to-one rule.
Extended · Syllabus E2.13
Notes and key methods
A function gives one output for every allowed input. Its domain is the input set and its range consists of the outputs it actually attains. For a strictly increasing linear rule on a closed interval, the endpoint images give the included range endpoints.
To find an inverse, write \(\displaystyle y=f(x)\), solve for x, then interchange input and output. A one-to-one rule has exactly one original input for each attained output. The inverse domain is the original range; its range is the original domain.
Check the conditions
Use the stated units and precision.
Show the method as well as the final answer.
Three worked examples
Example 1 · Functions, domains, ranges and linear inverses
\(f(x)=3 x + 4\) with domain \(\left[1, 5\right]\).
State the range of f and justify why it has an inverse.
Find the inverse rule and state its domain and range.
Find \(f^{-1}(10)\) and check it in f.
Find the gradient of the inverse rule.
Show the reasoning
(a)
The positive gradient 3 makes f strictly increasing. Its endpoint outputs are 7 and 19, and both are attained.
Answer: \(\left[7, 19\right]\); f is one-to-one on its stated domain.
(b)
Solve \(y=3 x + 4\) for x, interchange the variables, and exchange the original domain and range.