# 0.8 Exponential functions and graphs  (Page 2/2)

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

For example, the domain of $g\left(x\right)=3·{2}^{x+1}+2$ is $\left\{x:x\in \mathbb{R}\right\}$ . For the range,

$\begin{array}{ccc}\hfill {2}^{x+1}& >& 0\hfill \\ \hfill 3·{2}^{x+1}& >& 0\hfill \\ \hfill 3·{2}^{x+1}+2& >& 2\hfill \end{array}$

Therefore the range is $\left\{g\left(x\right):g\left(x\right)\in \left[2,\infty \right)\right\}$ .

## Domain and range

1. Give the domain of $y={3}^{x}$ .
2. What is the domain and range of $f\left(x\right)={2}^{x}$ ?
3. Determine the domain and range of $y={\left(1,5\right)}^{x+3}$ .

## Intercepts

For functions of the form, $y=a{b}^{\left(x+p\right)}+q$ , 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:

$\begin{array}{ccc}\hfill y& =& a{b}^{\left(x+p\right)}+q\hfill \\ \hfill {y}_{int}& =& a{b}^{\left(0+p\right)}+q\hfill \\ & =& a{b}^{p}+q\hfill \end{array}$

For example, the $y$ -intercept of $g\left(x\right)=3·{2}^{x+1}+2$ is given by setting $x=0$ to get:

$\begin{array}{ccc}\hfill y& =& 3·{2}^{x+1}+2\hfill \\ \hfill {y}_{int}& =& 3·{2}^{0+1}+2\hfill \\ & =& 3·{2}^{1}+2\hfill \\ & =& 3·2+2\hfill \\ & =& 8\hfill \end{array}$

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

$\begin{array}{ccc}\hfill y& =& a{b}^{\left(x+p\right)}+q\hfill \\ \hfill 0& =& a{b}^{\left({x}_{int}+p\right)}+q\hfill \\ \hfill a{b}^{\left({x}_{int}+p\right)}& =& -q\hfill \\ \hfill {b}^{\left({x}_{int}+p\right)}& =& -\frac{q}{a}\hfill \end{array}$

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 $y=a{b}^{\left(x+p\right)}+q$ does not have any $x$ -intercepts.

For example, the $x$ -intercept of $g\left(x\right)=3·{2}^{x+1}+2$ is given by setting $y=0$ to get:

$\begin{array}{ccc}\hfill y& =& 3·{2}^{x+1}+2\hfill \\ \hfill 0& =& 3·{2}^{{x}_{int}+1}+2\hfill \\ \hfill -2& =& 3·{2}^{{x}_{int}+1}\hfill \\ \hfill {2}^{{x}_{int}+1}& =& \frac{-2}{2}\hfill \end{array}$

which has no real solution. Therefore, the graph of $g\left(x\right)=3·{2}^{x+1}+2$ does not have a $x$ -intercept. You will notice that calculating $g\left(x\right)$ 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 $y=\frac{1}{2}{\left(1,5\right)}^{x+3}-0,75$ .

## Asymptotes

Functions of the form $y=a{b}^{\left(x+p\right)}+q$ always have exactly one horizontal asymptote.

When examining the range of these functions, we saw that we always have either $y 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 $g\left(x\right)=3·{2}^{x+1}+2$ is $\left(2,\infty \right)$ because $g\left(x\right)$ is always greater than 2. However, the value of $g\left(x\right)$ can get extremely close to 2, even though it never reaches it. For example, if you calculate $g\left(-20\right)$ , the value is approximately 2.000006. Using larger negative values of $x$ will make $g\left(x\right)$ even closer to 2: the value of $g\left(-100\right)$ 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 $y=3{\left(0,8\right)}^{x-1}-3$ ?

## Sketching graphs of the form $f\left(x\right)=a{b}^{\left(x+p\right)}+q$

In order to sketch graphs of functions of the form, $f\left(x\right)=a{b}^{\left(x+p\right)}+q$ , we need to determine four characteristics:

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

For example, sketch the graph of $g\left(x\right)=3·{2}^{x+1}+2$ . Mark the intercepts.

We have determined the domain to be $\left\{x:x\in \mathbb{R}\right\}$ and the range to be $\left\{g\left(x\right):g\left(x\right)\in \left(2,\infty \right)\right\}$ .

The $y$ -intercept is ${y}_{int}=8$ and there is no $x$ -intercept.

## 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 $f\left(x\right)={3}^{x}$ .
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 $f\left(x\right)=1+a.{2}^{x}$ (a is a constant) passes through the origin.
1. Determine the value of $a$ .
2. Determine the value of $f\left(-15\right)$ correct to FIVE decimal places.
3. Determine the value of $x$ , if $P\left(x;0,5\right)$ 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 $f\left(x\right)=a.{b}^{x}\phantom{\rule{3.33333pt}{0ex}}\left(a\ne 0\right)$ 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 $f\left(13\right)$ .
4. Describe the transformation of the curve of $f$ to $h$ if $h\left(x\right)=f\left(-x\right)$ .

how environment affect demand and supply of commodity ?
Wht at the criteria for market ?
Amos
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monetary policy is a policy thrust by National Govt(CBN) to influence government spending, purchase &taxes
Frank
necessity of economics
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
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Frank
price is different from demand- demand is amount of commodity
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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
show that shortrun average cost
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Mbah
what is money
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agaba
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Wesonga
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agaba
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MUDASIRU
what is the formulae for calculating national income
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classify the production units like agriculture, banking, transport etc
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definition of unemployment
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assani
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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.
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
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Emmanuel
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Emmanuel
I would like to bring in Educational levels can also be the cause the cause of the problem respectively
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ananti
lack of skills among the new generation is the serious issue.
Vishal
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Joe
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Can someone please define what economics is
economics simply is a social science subject that study human behavior.
dajan
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Alao
Can someone please tell me how to calculate GDP
Emmanuel
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Alao
thanks Alae
Emmanuel
u are welcome
Alao
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Joe
what is the law of demand
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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...
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Baraka
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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
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Difference between extinct and extici spicies
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
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Other chapter Q/A we can ask