Q31. - Algebra 2

Question

Determine whether the function $y={30}^{-x}$ represents exponential growth or decay.

Step-by-Step Solution

Answer

The given function $y={30}^{-x}$ represents exponential decay.

Step 1. Write down the given information.

The given function is $y={30}^{-x}$.

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;

$\begin{array}{l}y={30}^{-x}\\ y={\left({30}^{-1}\right)}^x\\ y={\left(\frac{1}{30}\right)}^x\end{array}$

From the concept stated above it can be interpreted that the function $y={\left(\frac{1}{30}\right)}^x$ will represent exponential decay because the base (b) of the function $y={\left(\frac{1}{30}\right)}^x$ is such that $0<b<1$. Hence, the given function $y={30}^{-x}$ represents exponential decay.

Step 4. Conclusion.

The given function $y={30}^{-x}$ represents exponential decay.