Question

The inverse function
f
−
1
(
x
)
can be formed if the domain of
f
(
x
)
is restricted. Which of the options best defines this restricted domain?


0
≤
x
≤
3.6


1
≤
x
≤
10


1
≤
x
≤
7.5


3.6
≤
x
≤
10

Answer 1

I don’t have the definition or graph of f(x) in your question, so I can’t pick the correct interval yet.

To decide which restricted domain makes f invertible you need an interval on which f is one-to-one (monotonic). Steps to do that:

1. Give me f(x) (or its graph).
2. I will find where f′(x)=0 or is undefined (critical points) and determine the sign of f′ on the subintervals.
3. Any option that lies entirely inside an interval where f′ keeps the same sign (always ≥0 or always ≤0, i.e. always increasing or always decreasing) is a valid restriction making f invertible. If more than one option is monotonic for f, choose the one the problem asks for (usually the largest convenient interval or the one covering the original domain of interest).

Paste the function or the graph and I’ll check the four choices and pick the correct one.