Learning Objectives
Introduction
Tutorial
Let f and g be two functions such that
f(g(x)) = x, for every x in the domain of g
g(f(x))
= x,
for
every x in the domain of f,
then the function g is said to be the inverse of the function f and is denoted .
The domain of f is equal to the range of and the range of f is the domain of .
Let’s look at f(g(x))
first:
Next, let’s look at g(f(x)):
Since f(g(x)) AND g(f(x)) BOTH came out to be x, this proves that the two functions are inverses of each other.
Let’s look at f(g(x))
first:
Since BOTH f(g(x)) AND g(f(x)) would have to equal x for them to be inverses of each other and f(g(x)) is not equal to x, then we can stop here and say without a doubt that they are NOT inverses of each other.
If NO horizontal line can be drawn so
that it intersects a
graph of a function f more than once, then the function f has an inverse
function
This graph would pass the horizontal
line test, because there
would not be any place on it that we could draw a horizontal line where
it would
intersect the graph of the function in more than one place.
Therefore, this function has an inverse function.
This function would not pass the
horizontal line test
because there is at least one place on it that we could draw a
horizontal line
and intersect it in more than one place. In fact, there are a lot
of
horizontal lines that we can draw that would intersect it in more than
one
place, but we only need to show one to say this function does not have
an
inverse function.
The graph below shows one horizontal
line drawn through our
graph that intersects it in two places:
If this y is a function, it is
the inverse of the original
function.
Since this y is a function, it
is the inverse of the original
function.
Since this y is a function, it
is the inverse of the original
function.
The graph of is a
reflection of the
graph of the function f about the line y = x.
The next two examples will illustrate
this.
Since this y is a function, it
is the inverse of the original
function.
See how the
graphs of f and are
mirrored images across the line y = x? Note how the ordered pairs are interchanged … for
example (2, 0) is a
point on f where (0, 2) is a point on
If you need a review on graphing linear functions, feel free to go to
College Algebra Tutorial
27: Graphing Lines.
The range of f is the same as
the domain of which is .
If you need a review on
finding the domain and range of a function, feel free to go to College
Algebra Tutorial
30: Introduction to Functions.
If you need a review on finding the domain and range of a graph of a function, feel free to go to College Algebra Tutorial 32: Graphs of Functions, Part II.
Since this y is a function, it
is the inverse of the original
function.
The range of f is the same as
the domain of which is .
If you need a review on
finding the domain and range of a function, feel free to go to College
Algebra Tutorial
30: Introduction to Functions.
If you need a review on finding the domain and range of a graph of a function, feel free to go to College Algebra Tutorial 32: Graphs of Functions, Part II.
Practice Problems
To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.
Practice Problems 1a - 1b: Use the definition of inverse functions to determine if the functions f and g are inverses of each other.
Practice Problems 2a - 2b: Given the function: a) find the equation of , b) graph f and and c) indicate the domain and range of f and using interval notation.
Need Extra Help on these Topics?
There are no appropriate webpages
that can assist
you in the topics that were covered on this page.
Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.
Last revised on June 24, 2010 by Kim Seward.
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