Question

determine whether each function represents exponential growth or exponential decay.
exponential growth
exponential decay
:: $y = \frac{1}{2}e^{-x}$
:: $f(x) = 0.25e^{3x}$
:: $f(x) = 5e^{-0.5x}$
:: $y = 8e^{0.1x}$

Explanation:

Step1: Recall growth/decay rule

An exponential function in the form $y = ae^{kx}$ represents exponential growth if $k>0$, and exponential decay if $k<0$.

Step2: Analyze $y=\frac{1}{2}e^{-x}$

Here, $k=-1<0$, so it is decay.

Step3: Analyze $f(x)=0.25e^{3x}$

Here, $k=3>0$, so it is growth.

Step4: Analyze $f(x)=5e^{-0.5x}$

Here, $k=-0.5<0$, so it is decay.

Step5: Analyze $y=8e^{0.1x}$

Here, $k=0.1>0$, so it is growth.

Answer:

Exponential Growth:

• $f(x)=0.25e^{3x}$
• $y=8e^{0.1x}$

Exponential Decay:

• $y=\frac{1}{2}e^{-x}$
• $f(x)=5e^{-0.5x}$