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

May 24, 2018

$f \left(x\right)$ is decreasing..

#### Explanation:

$f \left(x\right) = {\left(\frac{1}{5}\right)}^{x}$

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

${\left(\frac{1}{5}\right)}^{x} = \frac{{1}^{x}}{{5}^{x}}$

but 1 to any power is just 1 so:

${\left(\frac{1}{5}\right)}^{x} = \frac{{1}^{x}}{{5}^{x}} = \frac{1}{{5}^{x}}$

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

$f \left(1\right) = \frac{1}{5} = 0.2$

$f \left(2\right) = \frac{1}{25} = 0.04$

$f \left(3\right) = \frac{1}{125} = 0.008$

So $f \left(x\right)$ is decreasing closer and closer to 0.

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

May 24, 2018

Decreasing

#### Explanation:

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

In graphs of the form $f \left(x\right) = {a}^{x}$ 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 $f \left(x\right) = {\left(\frac{1}{5}\right)}^{x}$. 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!