Question

Is the following growth or decay f(x)=3(0.4)^x

Answer 1

We have the function
`f(x)=3(0.4)^(x)` and we want to know if the function is growing or decaying.
The first thing we need to do is convert the decimal into a fraction. To do this we are going to add the denominator 1 to the decimal, and then we'll multiply both numerator and denominator by a power of 10 for every number after the decimal point:

`(0.4)/(1) . (10)/(10) = (4)/(10) = (2)/(5)`
Now we can rewrite our function:

`f(x)=3( (2)/(5) )^(x)`
Lets replace
`x` with some integers to see how our function behaves; keep in mind that to raise a fraction to an exponent, we just raise both the numerator and denominator to the exponent:
-
`f(2)=3( (2)/(5) )^(2)`

`f(2)=3( (4)/(25))`

`f(x)=0.48`
-
`f(3)=3( (2)/(5) )^(3)`

`f(3)=3( (8)/(125) )`

`f(x)=0.192`
-
`f(5)=3( (2)/(5) )^(5)`

`f(5)=3( (32)/(3125) )`

`f(5)=0.03072`

Notice that the denominator is growing more faster than the denominator; therefore the function is decaying.