Page 55 - Calculus Demystified

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SOLUTION CHAPTER 1 Basics
First we double, then we add 3, then we square. So let f(x) = 2x, g(x) =
2
x + 3, h(x) = x . Then k(x) = (h ◦ g ◦ f )(x).
EXAMPLE 1.39
Write the function
2
r(t) = 2
t + 3
asthe composition of two functions.
SOLUTION
Firstwesquaret andadd3,thenwedivide2bythequantityjustobtained.Asa
2
result, we define f(t) = t +3 and g(t) = 2/t. It follows that r(t) = (g◦f )(t).
2
You Try It: Express the function g(x) = 3/(x + 5) as the composition of two
functions. Can you express it as the composition of three functions?

1.8.5 THE INVERSE OF A FUNCTION

Let f be the function which assigns to each working adult American his or her
Social Security Number (a 9-digit string of integers). Let g be the function which
assigns to each working adult American his or her age in years (an integer between
0 and 150). Both functions have the same domain, and both take values in the non-
negative integers. But there is a fundamental difference between f and g.Ifyou
are given a Social Security number, then you can determine the person to whom it
belongs. There will be one and only one person with that number. But if you are
given a number between 0 and 150, then there will probably be millions of people
with that age. You cannot identify a person by his/her age. In summary, if you know
g(x) then you generally cannot determine what x is. But if you know f(x) then you
can determine what (or who) x is. This leads to the main idea of this subsection.
Let f : S → T be a function. We say that f has an inverse (is invertible) if
there is a function f −1 : T → S such that (f ◦ f −1 )(t) = t for all t ∈ T and
(f −1 ◦f )(s) = s for all s ∈ S. Notice that the symbol f −1 denotes a new function
which we call the inverse of f .

Basic Rule forFinding Inverses To find the inverse of a function f ,we
solve the equation
(f ◦ f −1 )(t) = t
for the function f −1 (t).
EXAMPLE 1.40
Find the inverse of the function f(s) = 3s.