We have the function [tex]f(x)=3(0.4)^{x} [/tex] 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:
[tex] \frac{0.4}{1} . \frac{10}{10} = \frac{4}{10} = \frac{2}{5} [/tex]
Now we can rewrite our function:
[tex]f(x)=3( \frac{2}{5} )^{x} [/tex]
Lets replace [tex]x[/tex] 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:
-[tex]f(2)=3( \frac{2}{5} )^{2} [/tex]
[tex]f(2)=3( \frac{4}{25}) [/tex]
[tex]f(x)=0.48[/tex]
-[tex]f(3)=3( \frac{2}{5} )^{3} [/tex]
[tex]f(3)=3( \frac{8}{125} )[/tex]
[tex]f(x)=0.192[/tex]
-[tex]f(5)=3( \frac{2}{5} )^{5} [/tex]
[tex]f(5)=3( \frac{32}{3125} )[/tex]
[tex]f(5)=0.03072[/tex]
Notice that the denominator is growing more faster than the denominator; therefore the function is decaying.
