Question

Given the exponential function f(x) = 54(0.45)x, classify the function as exponential growth or decay and determine the percent rate of growth or decay.

A.Exponential decay, 55% decrease
B.Exponential growth, 45% increase
C.Exponential decay, 45% decrease
D.Exponential growth, 55% increase

Answer 1

Exponential decay, 55% decrease

Explanation:

`f(x) = 54(0.45)^x`

General exponential growth function is
`y=a(1+r)^x`

exponential growth function is
`y=a(1-r)^x`

The value of 1-r is less than 1 then it is exponential decay

In the given f(x) , the 1-r is 0.45 that is less than 1

So it is exponential decay.

`1-r= 0.45`

Subtract 1 on both sides

`r=0.55`

Multiply by 100 to get %

r= 55%

Exponential decay, 55% decrease

Answer 2

Option A.Exponential decay, 55% decrease

Explanation:

we have

`f(x)=54(0.45)^(x)`

This is a exponential function of the form

`f(x)=a(b)^(x)`

where

a is the initial value

b is the base

b=(1+r)

r is the rate of change

In this problem

a=54

b=0.45

so

0.45=1+r

r=0.45-1

r=-0.55

Convert to percentage

r=-55% ------> is negative because is a exponential decay