Answer:
f(x) = 0.25(5.25)ˣ
Step-by-step explanation:
In order to show exponential growth, the numbers must be positive and the base of the exponent must be larger than 1.
Think of it as a percentage. The base of the exponent will be the percentage of the original amount taken each time; if it is less than 1.00 then it would shrink. Larger than 1.00 will grow. The only one with positive numbers and a growth factor of greater than 1 is f(x) = 0.25(5.25)ˣ.
The exponential growth function is [tex]\boxed{f\left( x \right) = 0.25 \times {{\left( {5.25} \right)}^x}}[/tex]. Option (b) is correct.
Further explanation:
Given:
The options are as follows,
(a). [tex]f\left( x \right) = 6{\left( {0.25} \right)^x}[/tex]
(b). [tex]f\left( x \right) = 0.25{\left( {5.25} \right)^x}[/tex]
(c). [tex]f\left( x \right) = {\left( { - 4.25} \right)^x}[/tex]
(d). [tex]f\left( x \right) = {\left( { - 1.25} \right)^x}[/tex]
Explanation:
The general form of exponential function can be expressed as follows,
[tex]f\left( x \right) = a \times {\left( {b} \right)^x}[/tex]
Here, [tex]a[/tex] represents the initial value, [tex]f\left( x \right)[/tex] is the value after time [tex]x[/tex] and [tex]b[/tex] represents the growth rate.
For exponential grown function the value of b must be greater than 1.
In option (a) value of [tex]b[/tex] is less than 1. Therefore, it doesn’t represent the exponential growth.
In option (b) value of [tex]b[/tex] is greater than 1. Therefore, itrepresents the exponential growth.
In option (c) value of [tex]b[/tex] is less than 1. Therefore, it doesn’t represent the exponential growth.
In option (d) value of [tex]b[/tex] is less than 1. Therefore, it doesn’t represent the exponential growth.
The exponential growth function is [tex]\boxed{f\left( x \right) = 0.25 \times {{\left( {5.25} \right)}^x}}[/tex]. Option (b) is correct.
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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.
