<< Chapter < Page Chapter >> Page >

Intercepts

The general form of the inverse function of the form y = a x + q is y = 1 a x - q a .

By setting x = 0 we have that the y -intercept is y i n t = - q a . Similarly, by setting y = 0 we have that the x -intercept is x i n t = q .

It is interesting to note that if f ( x ) = a x + q , then f - 1 ( x ) = 1 a x - q a and the y -intercept of f ( x ) is the x -intercept of f - 1 ( x ) and the x -intercept of f ( x ) is the y -intercept of f - 1 ( x ) .

Exercises

  1. Given f ( x ) = 2 x - 3 , find f - 1 ( x )
  2. Consider the function f ( x ) = 3 x - 7 .
    1. Is the relation a function?
    2. If it is a function, identify the domain and range.
  3. Sketch the graph of the function f ( x ) = 3 x - 1 and its inverse on the same set of axes.
  4. The inverse of a function is f - 1 ( x ) = 2 x - 4 , what is the function f ( x ) ?

Inverse function of y = a x 2

The inverse relation, possibly a function, of y = a x 2 is determined by solving for x as:

y = a x 2 x 2 = y a x = ± y a
The function f ( x ) = x 2 and its inverse f - 1 ( x ) = ± x . The line y = x is shown as a dashed line.

We see that the inverse ”function” of y = a x 2 is not a function because it fails the vertical line test. If we draw a vertical line through the graph of f - 1 ( x ) = ± x , the line intersects the graph more than once. There has to be a restriction on the domain of a parabola for the inverse to also be a function. Consider the function f ( x ) = - x 2 + 9 . The inverse of f can be found by witing f ( y ) = x . Then

x = - y 2 + 9 y 2 = 9 - x y = ± 9 - x

If we restrict the domain of f ( x ) to be x 0 , then 9 - x is a function. If the restriction on the domain of f is x 0 then - 9 - x would be a function, inverse to f .

Khan academy video on inverse functions - 2

Khan academy video on inverse functions - 3

Exercises

  1. The graph of f - 1 is shown. Find the equation of f , given that the graph of f is a parabola. (Do not simplify your answer)
  2. f ( x ) = 2 x 2 .
    1. Draw the graph of f and state its domain and range.
    2. Find f - 1 and, if it exists, state the domain and range.
    3. What must the domain of f be, so that f - 1 is a function ?
  3. Sketch the graph of x = - 10 - y 2 . Label a point on the graph other than the intercepts with the axes.
    1. Sketch the graph of y = x 2 labelling a point other than the origin on your graph.
    2. Find the equation of the inverse of the above graph in the form y = ... .
    3. Now sketch the graph of y = x .
    4. The tangent to the graph of y = x at the point A(9;3) intersects the x -axis at B. Find the equation of this tangent and hence or otherwise prove that the y -axis bisects the straight line AB.
  4. Given: g ( x ) = - 1 + x , find the inverse of g ( x ) in the form g - 1 ( x ) = ... .

Inverse function of y = a x

The inverse function of y = a x 2 is determined by solving for x as follows:

y = a x log ( y ) = log ( a x ) = x log ( a ) x = log ( y ) log ( a )

The inverse of y = 10 x is x = 10 y , which we write as y = log ( x ) . Therefore, if f ( x ) = 10 x , then f - 1 = log ( x ) .

The function f ( x ) = 10 x and its inverse f - 1 ( x ) = log ( x ) . The line y = x is shown as a dashed line.

The exponential function and the logarithmic function are inverses of each other; the graph of the one is the graph of the other, reflected in the line y = x . The domain of the function is equal to the range of the inverse. The range of the function is equal to the domain of the inverse.

Exercises

  1. Given that f ( x ) = ( 1 5 ) x , sketch the graphs of f and f - 1 on the same system of axes indicating a point on each graph (other than the intercepts) and showing clearly which is f and which is f - 1 .
  2. Given that f ( x ) = 4 - x ,
    1. Sketch the graphs of f and f - 1 on the same system of axes indicating a point on each graph (other than the intercepts) and showing clearly which is f and which is f - 1 .
    2. Write f - 1 in the form y = ... .
  3. Given g ( x ) = - 1 + x , find the inverse of g ( x ) in the form g - 1 ( x ) = ...
  4. Answer the following questions:
    1. Sketch the graph of y = x 2 , labeling a point other than the origin on your graph.
    2. Find the equation of the inverse of the above graph in the form y = ...
    3. Now, sketch y = x .
    4. The tangent to the graph of y = x at the point A ( 9 ; 3 ) intersects the x -axis at B . Find the equation of this tangent, and hence, or otherwise, prove that the y -axis bisects the straight line A B .

End of chapter exercises

  1. Sketch the graph of x = - 10 - y 2 . Is this graph a function ? Verify your answer.
  2. f ( x ) = 1 x - 5 ,
    1. determine the y -intercept of f ( x )
    2. determine x if f ( x ) = - 1 .
  3. Below, you are given 3 graphs and 5 equations.
    1. y = log 3 x
    2. y = - log 3 x
    3. y = log 3 ( - x )
    4. y = 3 - x
    5. y = 3 x
    Write the equation that best describes each graph.
  4. Given g ( x ) = - 1 + x , find the inverse of g ( x ) in the form g - 1 ( x ) = ...
  5. Consider the equation h ( x ) = 3 x
    1. Write down the inverse in the form h - 1 ( x ) = . . .
    2. Sketch the graphs of h ( x ) and h - 1 ( x ) on the same set of axes, labelling the intercepts with the axes.
    3. For which values of x is h - 1 ( x ) undefined ?
    1. Sketch the graph of y = 2 x 2 + 1 , labelling a point other than the origin on your graph.
    2. Find the equation of the inverse of the above graph in the form y = ...
    3. Now, sketch y = x .
    4. The tangent to the graph of y = x at the point A ( 9 ; 3 ) intersects the x -axis at B . Find the equation of this tangent, and hence, or otherwise, prove that the y -axis bisects the straight line A B .

Questions & Answers

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
Almas
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
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 Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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