2.1. GENERAL CONSIDERATIONS 49

Example 2.1.10 Let f (t) = t2 for t ∈ [0,1] and let g(t) = t2 for t ∈ [0,3]. Then these aredifferent functions because they have different domains.

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

Definition 2.1.11 For any function f : D( f ) ⊆ X → Y, define the following setknown as 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.

Polynomials and rational functions are particularly easy functions to understand be-cause they do come from a simple formula.

Definition 2.1.12 A function f is a polynomial if

f (x) = anxn +an−1xn−1 + · · ·+a1x+a0

where the ai are real or complex numbers and n is a nonnegative integer. In this case thedegree of the polynomial, f (x) is n. Thus the degree of a polynomial is the largest exponentappearing on the variable.

f is a rational function if

f (x) =h(x)g(x)

where h and g are polynomials.

For example, f (x) = 3x5 +9x2 +7x+5 is a polynomial of degree 5 and

f (x)≡ 3x5 +9x2 +7x+5x4 +3x+ x+1

is a rational function.Note that in the case of a rational function, the domain of the function might not be all

of R. For example, if f (x) = x2+8x+1 , the domain of f would be all complex numbers not

equal to −1. Also, f (x) is not a function. It is a function evaluated at x. The name of thefunction is f . Another thing which is often done is to denote the function in terms of analgorithm like

x → x2 − x+12x+6

This signifies the function f such that

f (x) =x2 − x+1

2x+6.

Closely related to the definition of a function is the concept of the graph of a function.