<< Chapter < Page Chapter >> Page >

Graphs of inverse functions

In earlier grades, you studied various types of functions and understood the effect of various parameters in the general equation. In this section, we will consider inverse functions .

An inverse function is a function which "does the reverse" of a given function. More formally, if f is a function with domain X , then f - 1 is its inverse function if and only if for every x X we have:

f - 1 ( f ( x ) ) = = x

A simple way to think about this is that a function, say y = f ( x ) , gives you a y -value if you substitute an x -value into f ( x ) . The inverse function tells you tells you which x -value was used to get a particular y -value when you substitue the y -value into f - 1 ( x ) . There are some things which can complicate this for example, think about a sin function, there are many x -values that give you a peak as the function oscillates. This means that the inverse of a sin function would be tricky to define because if you substitute the peak y -value into it you won't know which of the x -values was used to get the peak.

y = f ( x ) we have a function y 1 = f ( x 1 ) we substitute a specific x-value into the function to get a specific y-value consider the inverse function x = f - 1 ( y ) x = f - 1 ( y ) substituting the specific y-value into the inverse should return the specific x-value = f - 1 ( y 1 ) = x 1

This works both ways, if we don't have any complications like in the case of the sin function, so we can write:

f - 1 ( f ( x ) ) = f ( f - 1 ( x ) ) = x

For example, if the function x 3 x + 2 is given, then its inverse function is x ( x - 2 ) 3 . This is usually written as:

f : x 3 x + 2 f - 1 : x ( x - 2 ) 3

The superscript "-1" is not an exponent.

If a function f has an inverse then f is said to be invertible.

If f is a real-valued function, then for f to have a valid inverse, it must pass the horizontal line test , that is a horizontal line y = k placed anywhere on the graph of f must pass through f exactly once for all real k .

It is possible to work around this condition, by defining a “multi-valued“ function as an inverse.

If one represents the function f graphically in a x y -coordinate system, the inverse function of the equation of a straight line, f - 1 , is the reflection of the graph of f across the line y = x .

Algebraically, one computes the inverse function of f by solving the equation

y = f ( x )

for x , and then exchanging y and x to get

y = f - 1 ( x )

Khan academy video on inverse functions - 1

Inverse function of y = a x + q

The inverse function of y = a x + q is determined by solving for x as:

y = a x + q a x = y - q x = y - q a = 1 a y - q a

Therefore the inverse of y = a x + q is y = 1 a x - q a .

The inverse function of a straight line is also a straight line, except for the case where the straight line is a perfectly horizontal line, in which case the inverse is undefined.

For example, the straight line equation given by y = 2 x - 3 has as inverse the function, y = 1 2 x + 3 2 . The graphs of these functions are shown in [link] . It can be seen that the two graphs are reflections of each other across the line y = x .

The graphs of the function f ( x ) = 2 x - 3 and its inverse f - 1 ( x ) = 1 2 x + 3 2 . The line y = x is shown as a dashed line.

Domain and range

We have seen that the domain of a function of the form y = a x + q is { x : x R } and the range is { y : y R } . Since the inverse function of a straight line is also a straight line, the inverse function will have the same domain and range as the original function.

Questions & Answers

Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Siyavula textbooks: grade 12 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11242/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Siyavula textbooks: grade 12 maths' conversation and receive update notifications?

Ask