Question

Determine whether the function $y=0.2{\left(5\right)}^{-x}$ represents exponential growth or decay.

Short answer

The given function $y=0.2{\left(5\right)}^{-x}$ representsexponential decay.

Step-by-step solution

Step 1. Write down the given information.

The given function is $y=0.2{\left(5\right)}^{-x}$.

Step 2. Concept used.

The exponential function is of the form $y=a{b}^x$ represents exponential growth if the base $b>1$ and exponential decay if the base $0<b<1$ for all $a>0$.

Step 3. Explanation.

The given function can be re-written as;

$\begin{array}{l}y=0.2{\left(5\right)}^{-x}\mathrm{....}\left(\text{Given}\right)\\ y=0.2{\left(5^{-1}\right)}^x\\ y=0.2{\left(\frac{1}{5}\right)}^x\end{array}$

Now, from the concept stated above it can be interpreted that the function $y=0.2{\left(\frac{1}{5}\right)}^x$ will represent exponential decay because the base (b) of the function $y=0.2{\left(\frac{1}{5}\right)}^x$ is such that $0<b<1$. Hence, the given function $y=0.2{\left(5\right)}^{-x}$ represents exponential decay.

Step 4. Conclusion.

The given function $y=0.2{\left(5\right)}^{-x}$ represents exponential decay.