Respuesta :
Answer:
The correct option is, i.e., [tex]f(x)=2(\frac{1}{6})^x[/tex].
Step-by-step explanation:
The exponential function is defined as
[tex]f(x)=ab^x[/tex]
Where, a represents the vertical stretch and compression and b is the growth factor.
If |a|>1, then it represents vertical stretch and if |a|<1, then it represents vertical compression.
If b>1, then the function is a growth function and If 0<b<1, then the function is a decay function.
Since we have to find the function that is a stretch of the exponential decay function, therefore the value of a>1 and value of b<1.
In option 1, a>1 and value of b>1. So, option 1 is incorrect.
In option 2, a<1 and value of b>1. So, option 2 is incorrect.
In option 3, a>1 and value of b<1. So, option 3 is correct.
In option 4, a<1 and value of b<1. So, option 4 is incorrect.
The exponential decay function is [tex]f\left( x \right) = \dfrac{1}{2}{\left( 6 \right)^x}.[/tex]
Further explanation:
The general form of exponential function is,
[tex]f\left( x \right) = a \times {\left( {b} \right)^x}[/tex]
For exponential decay function the value of b must be less than 1.
Given:
The options are as follows,
(a). [tex]f\left( x \right) = 2{\left( 6 \right)^x}[/tex]
(b). [tex]f\left( x \right) = \dfrac{1}{2}{\left( 6 \right)^x}[/tex]
(c). [tex]f\left( x \right) = \dfrac{1}{2}{\left( 6 \right)^x}[/tex]
(d). [tex]f\left( x \right) = \dfrac{1}{2}\left( {\dfrac{1}{6}} \right)[/tex]
Explanation:
In option (a)
The value of [tex]b[/tex] is greater than 1. Therefore, it doesn’t represent the exponential decay.
In option (b)
The value of [tex]b[/tex] is greater than 1. Therefore, it doesn’t represent the exponential decay.
In option (c)
The value of [tex]b[/tex] is less than 1. Therefore, itrepresents the exponential decay.
In option (d)
The function doesn’t represent the exponential function.
The exponential decay function is [tex]f\left( x \right) = \dfrac{1}{2}{\left( 6 \right)^x}.[/tex]
Learn more:
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Exponential function
Keywords: population grow, rate of growth, growth model, exponential growth, estimates, t years, exponential growth model, standard form, point slope form, exponential function.
