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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 can chip be made from sand
Eke Reply
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 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
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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
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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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