4.1 Exponential functions  (Page 2/16)

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Apparently, the difference between “the same percentage” and “the same amount” is quite significant. For exponential growth, over equal increments, the constant multiplicative rate of change resulted in doubling the output whenever the input increased by one. For linear growth, the constant additive rate of change over equal increments resulted in adding 2 to the output whenever the input was increased by one.

The general form of the exponential function is $\text{\hspace{0.17em}}f\left(x\right)=a{b}^{x},\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is any nonzero number, $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ is a positive real number not equal to 1.

• If $\text{\hspace{0.17em}}b>1,$ the function grows at a rate proportional to its size.
• If $\text{\hspace{0.17em}}0 the function decays at a rate proportional to its size.

Let’s look at the function $\text{\hspace{0.17em}}f\left(x\right)={2}^{x}\text{\hspace{0.17em}}$ from our example. We will create a table ( [link] ) to determine the corresponding outputs over an interval in the domain from $\text{\hspace{0.17em}}-3\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}3.$

 $x$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$ $f\left(x\right)={2}^{x}$ ${2}^{-3}=\frac{1}{8}$ ${2}^{-2}=\frac{1}{4}$ ${2}^{-1}=\frac{1}{2}$ ${2}^{0}=1$ ${2}^{1}=2$ ${2}^{2}=4$ ${2}^{3}=8$

Let us examine the graph of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ by plotting the ordered pairs we observe on the table in [link] , and then make a few observations.

Let’s define the behavior of the graph of the exponential function $\text{\hspace{0.17em}}f\left(x\right)={2}^{x}\text{\hspace{0.17em}}$ and highlight some its key characteristics.

• the domain is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$
• the range is $\text{\hspace{0.17em}}\left(0,\infty \right),$
• as $\text{\hspace{0.17em}}x\to \infty ,f\left(x\right)\to \infty ,$
• as $\text{\hspace{0.17em}}x\to -\infty ,f\left(x\right)\to 0,$
• $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ is always increasing,
• the graph of $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ will never touch the x -axis because base two raised to any exponent never has the result of zero.
• $\text{\hspace{0.17em}}y=0\text{\hspace{0.17em}}$ is the horizontal asymptote.
• the y -intercept is 1.

Exponential function

For any real number $\text{\hspace{0.17em}}x,$ an exponential function is a function with the form

$f\left(x\right)=a{b}^{x}$

where

• $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is the a non-zero real number called the initial value and
• $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ is any positive real number such that $\text{\hspace{0.17em}}b\ne 1.$
• The domain of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is all real numbers.
• The range of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is all positive real numbers if $\text{\hspace{0.17em}}a>0.$
• The range of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is all negative real numbers if $\text{\hspace{0.17em}}a<0.$
• The y -intercept is $\text{\hspace{0.17em}}\left(0,a\right),$ and the horizontal asymptote is $\text{\hspace{0.17em}}y=0.$

Identifying exponential functions

Which of the following equations are not exponential functions?

• $f\left(x\right)={4}^{3\left(x-2\right)}$
• $g\left(x\right)={x}^{3}$
• $h\left(x\right)={\left(\frac{1}{3}\right)}^{x}$
• $j\left(x\right)={\left(-2\right)}^{x}$

By definition, an exponential function has a constant as a base and an independent variable as an exponent. Thus, $\text{\hspace{0.17em}}g\left(x\right)={x}^{3}\text{\hspace{0.17em}}$ does not represent an exponential function because the base is an independent variable. In fact, $\text{\hspace{0.17em}}g\left(x\right)={x}^{3}\text{\hspace{0.17em}}$ is a power function.

Recall that the base b of an exponential function is always a positive constant, and $\text{\hspace{0.17em}}b\ne 1.\text{\hspace{0.17em}}$ Thus, $\text{\hspace{0.17em}}j\left(x\right)={\left(-2\right)}^{x}\text{\hspace{0.17em}}$ does not represent an exponential function because the base, $\text{\hspace{0.17em}}-2,$ is less than $\text{\hspace{0.17em}}0.$

Which of the following equations represent exponential functions?

• $f\left(x\right)=2{x}^{2}-3x+1$
• $g\left(x\right)={0.875}^{x}$
• $h\left(x\right)=1.75x+2$
• $j\left(x\right)={1095.6}^{-2x}$

$g\left(x\right)={0.875}^{x}\text{\hspace{0.17em}}$ and $j\left(x\right)={1095.6}^{-2x}\text{\hspace{0.17em}}$ represent exponential functions.

Evaluating exponential functions

Recall that the base of an exponential function must be a positive real number other than $\text{\hspace{0.17em}}1.$ Why do we limit the base $b\text{\hspace{0.17em}}$ to positive values? To ensure that the outputs will be real numbers. Observe what happens if the base is not positive:

• Let $\text{\hspace{0.17em}}b=-9\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=\frac{1}{2}.\text{\hspace{0.17em}}$ Then $\text{\hspace{0.17em}}f\left(x\right)=f\left(\frac{1}{2}\right)={\left(-9\right)}^{\frac{1}{2}}=\sqrt{-9},$ which is not a real number.

Why do we limit the base to positive values other than $1?$ Because base $1\text{\hspace{0.17em}}$ results in the constant function. Observe what happens if the base is $1:$

Questions & Answers

For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
what is the standard form if the focus is at (0,2) ?
Lorejean Reply
a²=4
Roy Reply

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