Step-by-step explanation:
Content
Functions and their inverses
We begin with a simple example.
Example
Let f(x)=2x and g(x)=x2.
Apply the function g to the number 3, and then apply f to the result:
g(3)=32andf(32)=3.
A similar thing happens if we first apply f and then apply g:
f(3)=6andg(6)=3.
It is clear that this will happen with any starting number. This is expressed as
f(g(x))g(f(x))=x,for all x=x,for all x.
The function f reverses the effect of g, and the function g reverses the effect of f. We say that f and g are inverses of each other.
As another example, we have
(x−−√3)3=xandx3−−√3=x,
for all real x. So the functions f(x)=x3 and g(x)=x−−√3 are inverses of each other.
If x≥0, then (x−−√)2=x and x2−−√=x. If x<0, then x−−√ is not defined. So the functions f(x)=x2 and g(x)=x−−√ are inverses of each other, but we need to be careful about domains. We will look at this more carefully later in this section.
Basics
In an earlier section of this module, we defined the composite of two functions h and g by (g∘h)(x)=g(h(x)).
Definitions
The zero function 0–:R→R is defined by 0–(x)=0, for all x.
The identity function id:R→R is defined by id(x)=x, for all x.
Example
Consider a function f:R→R.
Prove that
0–∘f=0–
f∘id=f
id∘f=f.
Show that f∘0– does not necessarily equal 0–.
Solution
We have (0–∘f)(x)=0–(f(x))=0, for all x, and so 0–∘f=0–.
We have (f∘id)(x)=f(id(x))=f(x), for all x, and so f∘id=f.
We have (id∘f)(x)=id(f(x))=f(x), for all x, and so id∘f=f.
Consider the function given by f(x)=2, for all x. Then f∘0–(x)=f(0–(x))=f(0)=2, and so f∘0–≠0–.
Definition
Let f be a function with both domain and range all real numbers. Then the function g is the inverse of f if
f(g(x))g(f(x))=x,for all x,and=x,for all x.
That is, f∘g=id and g∘f=id.
Notes.
Clearly, if g is the inverse of f, then f is the inverse of g.
We denote the inverse of f by f−1. We read f−1 as 'f inverse'. Note that f inverse has nothing to do with the function 1f.
Example
Let f(x)=x+2 and let g(x)=x−2. Show that f and g are inverses of each other.
Solution
We have
f(g(x))=f(x−2)=x−2+2=x,for all x(f∘g=id)
and
g(f(x))=g(x+2)=x+2−2=x,for all x(g∘f=id).
Hence, the functions f and g are inverses of each other.
Exercise 5
Find the inverse of
f(x)=x+7
f(x)=4x+5.
Example
Let f(x)=ax+b with a≠0. Find the inverse of f.
Solution
We have x=f(x)−ba, for all x. So let g(x)=x−ba. Then
f(g(x))g(f(x))=f(x−ba)=a(x−ba)+b=x=g(ax+b)=(ax+b)−ba=x,
for all x. Hence, g is the inverse of f.
Exercise 6
Show that f(x)=x5 and g(x)=x15 are inverses of each other.
Find the inverse of f(x)=x3+2.
We do not yet have a general enough concept of inverses, since x2 and x−−√ do not fit into this framework, nor do ex and logex. We will give a definition that covers these functions later in this section.
The horizontal-line test
Consider the function f(x)=x2, which has domain the reals and range A={x:x≥0}. Does f have an inverse?
The following graph shows that it does not. We have f(−2)=f(2)=4, and so f−1(4) would have to take two values, −2 and 2! Hence, f does not have an inverse.
Graph of y = x squared and the line y = 4 on the one set of axes.
This idea can be formulated as a test.
Horizontal-line test
Let f be a function. If there is a horizontal line y=c that meets the graph y=f(x) at more than one point, then f does not have an inverse.
Notes. Remember that the vertical-line test determines whether a relation is a function.
Example
Consider the function
f(x)=x3−x=(x+1)x(x−1).
Its graph is shown in the following diagram.
Graph of y = x cubed minus x.
Does f have an inverse?
Solution
The line y=0 meets the graph at three points. By the horizontal-line test, the function f does not have an inverse.
