Question

Which is an exponential decay function?

Answer 1

The function which is an exponential decay function is:

`f(x)=(3)/(2)((8)/(7))^(-x)`

Explanation:

We know that an exponential function is in the form of:

`f(x)=ab^x`

where a>0 and if 0<b<1 then the function is a exponential decay function.

and if b>1 then the function is a exponential growth function.

a)

`f(x)=(3)/(4)((7)/(4))^x`

Here

`b=(7)/(4)>1`

Hence, the function is a exponential growth function.

b)

`f(x)=(2)/(3)((4)/(5))^(-x)`

We know that:

`a^(-x)=((1)/(a))^x`

Hence, we have the function f(x) as:

`f(x)=(2)/(3)((5)/(4))^x`

Here

`b=(5)/(4)>1`

Hence, the function is a exponential growth function.

c)

`f(x)=(3)/(2)((8)/(7))^(-x)`

We know that:

`a^(-x)=((1)/(a))^x`

Hence, we have the function f(x) as:

`f(x)=(3)/(2)((7)/(8))^x`

Here

`b=(7)/(8)<1`

Hence, the function is a exponential decay function.

d)

`f(x)=(1)/(3)((9)/(2))^x`

Here

`b=(9)/(2)>1`

Hence, the function is a exponential growth function.

Answer 2

Explanation:

exponential decay functions are written in the form :

`y=ab^(x)`

where b is less than 1

if we look at the 3rd choice and consider the term on the right.

`(8/7)^(-x)`

=
`(7/8)^(x)`

If we compare this to the general form above,

b = 7/8 (which is less than 1)

hence the 3rd choice is correct.