Download Definition of Inverse Functions

Definition:
Let f and g be two functions. If
f(g(x)) = x and g(f(x)) = x,
then g is the inverse of f and f is the inverse of g.
Graphs of Inverse Functions
We have seen examples of reflections in the plane. The reflection of a point (a,b) about the xaxis is (a,-b), and the reflection of (a,b) about the y-axis is (-a,b). Now we want to reflect about
the line y = x.
The reflection of the point (a,b) about the line y = x is the point (b,a).
Let f(x) = x3 + 2. Then f(2) = 10 and the point (2,10) is on the graph of f. The inverse of f must
take 10 back to 2, i.e. f-1(10)=2, so the point (10,2) is on the graph of f-1. The point (10,2) is the
reflection in the line y = x of the point (2,10). The same argument can be made for all points on
the graphs of f and f-1.
The graph of f-1 is the reflection about the line y = x of the graph of f.
Existence of an Inverse
Some functions do not have inverse functions. For example, consider f(x) = x2. There are two
numbers that f takes to 4, f(2) = 4 and f(-2) = 4. If f had an inverse, then the fact that f(2) = 4
would imply that the inverse of f takes 4 back to 2. On the other hand, since f(-2) = 4, the inverse
of f would have to take 4 to -2. Therefore, there is no function that is the inverse of f.
Look at the same problem in terms of graphs. If f had an inverse, then its graph would be the
reflection of the graph of f about the line y = x. The graph of f and its reflection about y = x are
drawn below.
Note that the reflected graph does not pass the vertical line test, so it is not the graph of a
function.
Vertical Line Test
A set of points in the plane is the graph of a function if and only if no vertical line intersects the
graph in more than one point.
Example 3.
The graph of the equation y2 = x + 5 is shown below.
By the vertical line test, this graph is not the graph of a function, because there are many vertical
lines that hit it more than once.
Think of the vertical line test this way. The points on the graph of a function f have the form (x,
f(x)), so once you know the first coordinate, the second is determined. Therefore, there cannot be
two points on the graph of a function with the same first coordinate.
All the points on a vertical line have the same first coordinate, so if a vertical line hits a graph
twice, then there are two points on the graph with the same first coordinate. If that happens, the
graph is not the graph of a function
Characteristics of Graphs
Consider the function f(x) = 2 x + 1. We recognize the equation y = 2 x + 1 as the SlopeIntercept form of the equation of a line with slope 2 and y-intercept (0,1).
Think of a point moving on the graph of f. As the point moves toward the right it rises. This is
what it means for a function to be increasing. Your text has a more precise definition, but this is
the basic idea.
The function f above is increasing everywhere. In general, there are intervals where a function is
increasing and intervals where it is decreasing.
The function graphed above is decreasing for x between -3 and 2. It is increasing for x less than 3 and for x greater than 2.
This generalizes as follows: A function f has an inverse if and only if when its graph is reflected
about the line y = x, the result is the graph of a function (passes the vertical line test). But this
can be simplified. We can tell before we reflect the graph whether or not any vertical line will
intersect more than once by looking at how horizontal lines intersect the original graph!
Horizontal Line Test
Let f be a function.
If any horizontal line intersects the graph of f more than once, then f does not have an inverse.
If no horizontal line intersects the graph of f more than once, then f does have an inverse.
The property of having an inverse is very important in mathematics, and it has a name.
Definition: A function f is one-to-one if and only if f has an inverse.
The following definition is equivalent, and it is the one most commonly given for one-to-one.
Alternate Definition: A function f is one-to-one if, for every a and b in its domain, f(a) = f(b)
implies a = b.
Exercise 2:
Graph the following functions and determine whether or not they have inverses.
(a) f(x) = (x - 3) x2.
(b) f(x) = x3 + 3x2 +3x.
(c) f(x) = x ^(1/3) ( the cube root of x).
A)
There is a horizontal line that hits the graph more than once.
f is not one-to-one.
f does not have an inverse.
B)
No horizontal line hits the graph more than once.
f is one-to-one.
f does have an inverse.
C)
No horizontal line hits the graph more than once.
f is one-to-one.
f does have an inverse.