54 CHAPTER 4. FUNCTIONS AND SEQUENCES

You should note that f (x) is not a function. It is the value of the function at the point x.The name of the function is f . Nevertheless, people often write f (x) to denote a functionand it doesn’t cause too many problems in beginning courses. When this is done, thevariable x should be considered as a generic variable free to be anything in D( f ) .

Sometimes people get hung up on formulas and think that the only functions of impor-tance are those which are given by some simple formula. It is a mistake to think this way.Functions involve a domain and a range and a function is determined by what it does. Thisis an old idea. See Luke 6:44 where Jesus says that you know a tree by its fruit. See alsoMatt. 7 about how to recognize false prophets. You look at what it does to determine whatit is. As it is with false prophets and trees, so it is with functions. 1 Although the idea isvery old, its application to mathematics started with Dirichlet2 in the early 1800’s becausehe was concerned with piecewise continuous functions which would be given by differentdescriptions on different intervals. Before his time, they did tend to think of functions interms of formulas.

Example 4.1.7 Let f (t) = t and g(t) = 1 + t. Then f g : R→ R is given by f g(t) =t (1+ t) = t + t2.

Example 4.1.8 Let f (t) = 2t +1 and g(t) =√

1+ t. Then

g◦ f (t) =√

1+(2t +1) =√

2t +2

for t ≥ −1. If t < −1 the inside of the square root sign is negative so makes no sense.Therefore, g◦ f : {t ∈ R : t ≥−1}→ R.

Note that in this last example, it was necessary to fuss about the domain of g◦ f becauseg is only defined for certain values of t.

The concept of a one to one function is very important. This is discussed in the follow-ing definition.

Definition 4.1.9 For any function f : D( f )⊆X→Y, define the following set knownas the inverse image of y.

f−1 (y)≡ {x ∈ D( f ) : f (x) = y} .

There may be many elements in this set, but when there is always only one element in thisset for all y∈ f (D( f )) , the function f is one to one sometimes written, 1−1. Thus f is oneto one, 1−1, if whenever f (x) = f (x1) , then x = x1. If f is one to one, the inverse functionf−1 is defined on f (D( f )) and f−1 (y) = x where f (x) = y. Thus from the definition,f−1 ( f (x)) = x for all x ∈ D( f ) and f

(f−1 (y)

)= y for all y ∈ f (D( f )) . Defining id by

id(z)≡ z this says f ◦ f−1 = id and f−1 ◦ f = id . Note that this is sloppy notation becausethe two id are totally different functions.

1In many religions, including mine, epistemology based on knowledge of good and evil as urged by Jesus,along with known facts, is disparaged and replaced with other criteria like feelings and emotions, proof texts ofscripture taken out of context, traditions, sacrifice of believers, claimed miracles, authority, and social pressure.

2Peter Gustav Lejeune Dirichlet, 1805-1859 was a German mathematician who did fundamental work inanalytic number theory. He also gave the first proof that Fourier series tend to converge to the mid-point of thejump of the function. He is a very important figure in the development of analysis in the nineteenth century. Aninteresting personal fact is that the great composer Felix Mendelsson was his brother in law.