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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 f ( x ) = a b x , where a is any nonzero number, b is a positive real number not equal to 1.

  • If b > 1 , the function grows at a rate proportional to its size.
  • If 0 < b < 1 , the function decays at a rate proportional to its size.

Let’s look at the function f ( x ) = 2 x from our example. We will create a table ( [link] ) to determine the corresponding outputs over an interval in the domain from 3 to 3.

x 3 2 1 0 1 2 3
f ( x ) = 2 x 2 3 = 1 8 2 2 = 1 4 2 1 = 1 2 2 0 = 1 2 1 = 2 2 2 = 4 2 3 = 8

Let us examine the graph of f by plotting the ordered pairs we observe on the table in [link] , and then make a few observations.

Graph of Companies A and B’s functions, which values are found in the previous table.

Let’s define the behavior of the graph of the exponential function f ( x ) = 2 x and highlight some its key characteristics.

  • the domain is ( , ) ,
  • the range is ( 0 , ) ,
  • as x , f ( x ) ,
  • as x , f ( x ) 0 ,
  • f ( x ) is always increasing,
  • the graph of f ( x ) will never touch the x -axis because base two raised to any exponent never has the result of zero.
  • y = 0 is the horizontal asymptote.
  • the y -intercept is 1.

Exponential function

For any real number x , an exponential function is a function with the form

f ( x ) = a b x

where

  • a is the a non-zero real number called the initial value and
  • b is any positive real number such that b 1.
  • The domain of f is all real numbers.
  • The range of f is all positive real numbers if a > 0.
  • The range of f is all negative real numbers if a < 0.
  • The y -intercept is ( 0 , a ) , and the horizontal asymptote is y = 0.

Identifying exponential functions

Which of the following equations are not exponential functions?

  • f ( x ) = 4 3 ( x 2 )
  • g ( x ) = x 3
  • h ( x ) = ( 1 3 ) x
  • j ( x ) = ( 2 ) x

By definition, an exponential function has a constant as a base and an independent variable as an exponent. Thus, g ( x ) = x 3 does not represent an exponential function because the base is an independent variable. In fact, g ( x ) = x 3 is a power function.

Recall that the base b of an exponential function is always a positive constant, and b 1. Thus, j ( x ) = ( −2 ) x does not represent an exponential function because the base, −2 , is less than 0.

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Which of the following equations represent exponential functions?

  • f ( x ) = 2 x 2 3 x + 1
  • g ( x ) = 0.875 x
  • h ( x ) = 1.75 x + 2
  • j ( x ) = 1095.6 2 x

g ( x ) = 0.875 x and j ( x ) = 1095.6 2 x represent exponential functions.

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Evaluating exponential functions

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

  • Let b = 9 and x = 1 2 . Then f ( x ) = f ( 1 2 ) = ( 9 ) 1 2 = 9 , which is not a real number.

Why do we limit the base to positive values other than 1 ? Because base 1 results in the constant function. Observe what happens if the base is 1 :

Questions & Answers

show that the set of all natural number form semi group under the composition of addition
Nikhil Reply
explain and give four Example hyperbolic function
Lukman Reply
_3_2_1
felecia
⅗ ⅔½
felecia
_½+⅔-¾
felecia
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
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Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
x*x=2
felecia
2+2x=
felecia
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
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Shirley Reply
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Abdullahi
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Mark
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
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Mark
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
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Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
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Seidu
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Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
Practice Key Terms 4

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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