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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

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
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 to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
Other chapter Q/A we can ask
Moahammedashifali Reply

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Source:  OpenStax, Siyavula textbooks: grade 11 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11243/1.3
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