Page 56 - Calculus Demystified

CHAPTER 1
SOLUTION Basics 43
We solve the equation
(f ◦ f −1 )(t) = t.

This is the same as
f(f −1 (t)) = t.

We can rewrite the last line as
3 · f −1 (t) = t
or
t
f −1 (t) = .
3
Thus f −1 (t) = t/3.

EXAMPLE 1.41
5
Let f : R → R be defined by f(s) = 3s . Find f −1 .
SOLUTION
We solve
(f ◦ f −1 )(t) = t

or
f(f −1 (t)) = t

or
5
3[f −1 (t)] = t
or
t
5
[f −1 (t)] =
3
or
1/5
t
f −1 (t) = .
3
√
You Try It: Find the inverse of the function g(x) = 3 x − 5.
It is important to understand that some functions do not have inverses.
EXAMPLE 1.42
2
Let f : R →{t : t ≥ 0} be defined by f(s) = s . If possible, find f −1 .