<< Chapter < Page Chapter >> Page >

Therefore, if a < 0 , then the range is ( - , q ) , meaning that f ( x ) can be any real number less than q . Equivalently, one could write that the range is { y R : y < q } .

For example, the domain of g ( x ) = 3 · 2 x + 1 + 2 is { x : x R } . For the range,

2 x + 1 > 0 3 · 2 x + 1 > 0 3 · 2 x + 1 + 2 > 2

Therefore the range is { g ( x ) : g ( x ) [ 2 , ) } .

Domain and range

  1. Give the domain of y = 3 x .
  2. What is the domain and range of f ( x ) = 2 x ?
  3. Determine the domain and range of y = ( 1 , 5 ) x + 3 .

Intercepts

For functions of the form, y = a b ( x + p ) + q , the intercepts with the x - and y -axis are calculated by setting x = 0 for the y -intercept and by setting y = 0 for the x -intercept.

The y -intercept is calculated as follows:

y = a b ( x + p ) + q y i n t = a b ( 0 + p ) + q = a b p + q

For example, the y -intercept of g ( x ) = 3 · 2 x + 1 + 2 is given by setting x = 0 to get:

y = 3 · 2 x + 1 + 2 y i n t = 3 · 2 0 + 1 + 2 = 3 · 2 1 + 2 = 3 · 2 + 2 = 8

The x -intercepts are calculated by setting y = 0 as follows:

y = a b ( x + p ) + q 0 = a b ( x i n t + p ) + q a b ( x i n t + p ) = - q b ( x i n t + p ) = - q a

Since b > 0 (this is a requirement in the original definition) and a positive number raised to any power is always positive, the last equation above only has a real solution if either a < 0 or q < 0 (but not both). Additionally, a must not be zero for the division to be valid. If these conditions are not satisfied, the graph of the function of the form y = a b ( x + p ) + q does not have any x -intercepts.

For example, the x -intercept of g ( x ) = 3 · 2 x + 1 + 2 is given by setting y = 0 to get:

y = 3 · 2 x + 1 + 2 0 = 3 · 2 x i n t + 1 + 2 - 2 = 3 · 2 x i n t + 1 2 x i n t + 1 = - 2 2

which has no real solution. Therefore, the graph of g ( x ) = 3 · 2 x + 1 + 2 does not have a x -intercept. You will notice that calculating g ( x ) for any value of x will always give a positive number, meaning that y will never be zero and so the graph will never intersect the x -axis.

Intercepts

  1. Give the y-intercept of the graph of y = b x + 2 .
  2. Give the x- and y-intercepts of the graph of y = 1 2 ( 1 , 5 ) x + 3 - 0 , 75 .

Asymptotes

Functions of the form y = a b ( x + p ) + q always have exactly one horizontal asymptote.

When examining the range of these functions, we saw that we always have either y < q or y > q for all input values of x . Therefore the line y = q is an asymptote.

For example, we saw earlier that the range of g ( x ) = 3 · 2 x + 1 + 2 is ( 2 , ) because g ( x ) is always greater than 2. However, the value of g ( x ) can get extremely close to 2, even though it never reaches it. For example, if you calculate g ( - 2 0 ) , the value is approximately 2.000006. Using larger negative values of x will make g ( x ) even closer to 2: the value of g ( - 1 0 0 ) is so close to 2 that the calculator is not precise enough to know the difference, and will (incorrectly) show you that it is equal to exactly 2.

From this we deduce that the line y = 2 is an asymptote.

Asymptotes

  1. Give the equation of the asymptote of the graph of y = 3 x - 2 .
  2. What is the equation of the horizontal asymptote of the graph of y = 3 ( 0 , 8 ) x - 1 - 3 ?

Sketching graphs of the form f ( x ) = a b ( x + p ) + q

In order to sketch graphs of functions of the form, f ( x ) = a b ( x + p ) + q , we need to determine four characteristics:

  1. domain and range
  2. y -intercept
  3. x -intercept

For example, sketch the graph of g ( x ) = 3 · 2 x + 1 + 2 . Mark the intercepts.

We have determined the domain to be { x : x R } and the range to be { g ( x ) : g ( x ) ( 2 , ) } .

The y -intercept is y i n t = 8 and there is no x -intercept.

Graph of g ( x ) = 3 · 2 x + 1 + 2 .

Sketching graphs

  1. Draw the graphs of the following on the same set of axes. Label the horizontal asymptotes and y-intercepts clearly.
    1. y = b x + 2
    2. y = b x + 2
    3. y = 2 b x
    4. y = 2 b x + 2 + 2
    1. Draw the graph of f ( x ) = 3 x .
    2. Explain where a solution of 3 x = 5 can be read off the graph.

End of chapter exercises

  1. The following table of values has columns giving the y -values for the graph y = a x , y = a x + 1 and y = a x + 1 . Match a graph to a column.
    x A B C
    -2 7,25 6,25 2,5
    -1 3,5 2,5 1
    0 2 1 0,4
    1 1,4 0,4 0,16
    2 1,16 0,16 0,064
  2. The graph of f ( x ) = 1 + a . 2 x (a is a constant) passes through the origin.
    1. Determine the value of a .
    2. Determine the value of f ( - 15 ) correct to FIVE decimal places.
    3. Determine the value of x , if P ( x ; 0 , 5 ) lies on the graph of f .
    4. If the graph of f is shifted 2 units to the right to give the function h , write down the equation of h .
  3. The graph of f ( x ) = a . b x ( a 0 ) has the point P(2;144) on f .
    1. If b = 0 , 75 , calculate the value of a .
    2. Hence write down the equation of f .
    3. Determine, correct to TWO decimal places, the value of f ( 13 ) .
    4. Describe the transformation of the curve of f to h if h ( x ) = f ( - x ) .

Questions & Answers

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
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
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
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
Other chapter Q/A we can ask
Moahammedashifali Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Siyavula textbooks: grade 11 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11243/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Siyavula textbooks: grade 11 maths' conversation and receive update notifications?

Ask