Question

Is the function `f(x) = (1/5)^x` increasing or decreasing?

Answer 1

`f(x)` is decreasing..

Explanation:

Let's think about this, the function is:

`f(x) = (1/5)^x`

so a fraction is being raised to a power, what does that mean?

`(1/5)^x = (1^x)/(5^x)`

but 1 to any power is just 1 so:

`(1/5)^x = (1^x)/(5^x) = (1)/(5^x)`

so as x gets bigger and bigger the number dividing 1 gets huge and the value gets closer and closer to 0.

`f(1) = 1/5 = 0.2`

`f(2) = 1/25 = 0.04`

`f(3) = 1/125 = 0.008`

So `f(x)` is decreasing closer and closer to 0.

graph{ (1/5)^x [-28.87, 28.87, -14.43, 14.44]}

Answer 2

Decreasing

Explanation:

graph{(1/5)^x [-20, 20, -10.42, 10.42]}

In graphs of the form `f(x)=a^x` where `0 < a<1`, as `x` increases, `y` decreases, and vice-versa.

As exponential decay is measured as when a population or group of something is declining, and the amount that decreases is proportional to the size of the population, we can clearly see that happening in the equation of `f(x)=(1/5)^x`. Also keep in mind that exponential decay relates to a proportional decrease in the positive direction of the `x`-axis, while exponential growth relates to a proportional increase in the positive direction of the `x`-axis, so just from looking at the graph the answer can be clearly seen.

I hope I helped!