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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 environment affect demand and supply of commodity ?
Amos Reply
Wht at the criteria for market ?
Amos
what is difference between monitory policy and fiscal policy?
Malik Reply
monetary policy is a policy thrust by National Govt(CBN) to influence government spending, purchase &taxes
Frank
necessity of economics
Pamela Reply
I will say want,choice,opportunity cost,scarcity,scale of preference
Alao
what is monopoly market.How price output are determined under monopoly market
bisham
b) Monopoly market is an impecfect market where s single firm having the innovation to produce a particular commodity.Prices are determined through output since there are no other competitive.
Frank
Monopoly market:firm has market power & does not respond to market price
Frank
Explain the process of price determination under perfect competition market with suitable diagram
bisham Reply
Price determination under perfect competition via this process :firms have no market power to influence price rather firms respond to market price.
Frank
price is different from demand- demand is amount of commodity
Effah Reply
demand is amount /quantity of commodity a potential buyer is willing to buy at a given price at market
Frank
demand is a desire of customer on commodity with the ability to pay it and willing to buy it at given price of commodity
Harika
demand is price of what
Faith Reply
show that shortrun average cost
Baby Reply
what is economics
Mbah Reply
what is money
Mbah
what is money
Mbah
Difine macro economics
agaba
money is a medium of exchange between goods and services,maybe inform of currency.
Wesonga
Economics is study of how human beings strive to satisfy numerous wants using limited available resources.
Wesonga
how do you find the maximum number of workers the firms should employ order to produce where there are increasing returns
Jane
what are implications of computing national income?.
agaba
pl
MUDASIRU
what is the formulae for calculating national income
MUDASIRU
it calculated by value added method
Praveen
classify the production units like agriculture, banking, transport etc
Praveen
money is anything that is generally acceptetable for human
Ogbaji
Estimate the net value added(NVA) at fixed cost by each industrial structure
Praveen
definition of unemployment
Adam Reply
what are the causes of unemployment?
Mbubi Reply
The main causes of unemployment are listed below. 1. Frictional unemployment 2. Cyclical unemployment 3. Structural unemployment
assani
We can also categorize the causes on a broader sense as: 1. Political and 2. Social cause As unemployeement root causes are embaded in this two.
Yonathan
would opportunity cost exist if there was no scarcity?
assani
yes just because the opportunity cost arose when there is Alternative to choose among the alternatives.
BADAMASIU
I am thinking that, if our resources were unlimited, then there wouldn't be any need to forgo some wants. Hence the inexistence if opportunity cost
assani
Politics
Job
politics has done what?
assani
consider time assani
Mary
I'm Emmanuel,...I taught the main cause is the change in gov't.
Emmanuel
...Lack of capital to set up a firm respectively
Emmanuel
🙈
Emmanuel
I would like to bring in Educational levels can also be the cause the cause of the problem respectively
Emmanuel
I think the main causes of unemployment is lack of INFRASTRUCTURAL DEVELOPMENT OVER POPULATION OVER DEPENDENT ON GOVERNMENT LACK OF SELF EMPOWERMENT...
ananti
lack of skills among the new generation is the serious issue.
Vishal
Where I come from , I don't see why education or personal aspects seem to do with unimployment, technically the motivation and eigerness in all works of live is there , dispite the cultural influence and physical bearriors;the thing we lacking is Government Support and open market ethics.
Joe
sorry about that-(repation). We have a over powering ethical political system that's displacing the marketing asspects of economy and causing large scale unemployment right across the board...
Joe
can someone Explain Expansionary Monetary Policy and Contractionary Monetary Policy Using one of the instrument of Monetary Policy? Please am kinda lost here?. ta
Emmanuel Reply
using a graph show the case of substitute and compliment goods
Ade Reply
can anyone give me a simple explanation to Five Sector Macroeconomics?
Emmanuel
Can someone please define what economics is
jason Reply
economics simply is a social science subject that study human behavior.
dajan
economics is a social science which studies human behaviour as a relationship between ends and scarce means that has alternative uses
Alao
Can someone please tell me how to calculate GDP
Emmanuel
emmanual kapal to calculate GDP (Gross Domestic Product) has three method in calculating it (1)income approach (2) expenditure approach (3) value added method
Alao
thanks Alae
Emmanuel
u are welcome
Alao
in basic terms economics is revered to as battery system, it date back to when Men sees the need to exchange sapless goods and produce to gain , either wealth , basic necessities or to establish trading ties for personal benefit or social asspects in terms of coexistence and continuity, future .
Joe
what is the law of demand
Berlinda Reply
keep other thing constant, when the price increases demand decrease when the price decreases demand increases of the commodity.
sj
all things being equal,quantity demanded decrease as price increase and increase as price decrease
Seth
there's practial joke to it ..." the higher the demand ; scarcity, increase in production and drop in quality"... quite the controversy - for example China vs Europe, United States and we are all boxed up in between somewhere...
Joe
Other thing remain constant the low price of commodity the high quantity of commodity and vice versa is true
Baraka
Explain Effective demand
Anita Reply
What is effective demand
Anita
like Modi is in demand...best example of effective demand
Pranav
Don't get you
Anita
Anita you mean you don't get me or who?
Onyeking
level of demand that represents a real intention to purchase by people with the means to pay
Pranav
Difference between extinct and extici spicies
Amanpreet Reply
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
Beverly Reply
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
Mapo Reply
The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
Mapo Reply
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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