which of the following functions will decrease most rapidly?\n\na. $f(x) = 2 \\cdot (\\frac{1}{2})^x$\n\nb…

which of the following functions will decrease most rapidly?\n\na. $f(x) = 2 \\cdot (\\frac{1}{2})^x$\n\nb. $f(x) = 5 \\cdot (\\frac{3}{4})^x$\n\nc. $f(x) = 3 \\cdot (\\frac{9}{10})^x$\n\nd. $f(x) = 4 \\cdot (\\frac{5}{6})^x$

which of the following functions will decrease most rapidly?\n\na. $f(x) = 2 \\cdot (\\frac{1}{2})^x$\n\nb. $f(x) = 5 \\cdot (\\frac{3}{4})^x$\n\nc. $f(x) = 3 \\cdot (\\frac{9}{10})^x$\n\nd. $f(x) = 4 \\cdot (\\frac{5}{6})^x$

Answer

Explanation:

Step1: Identify the general exponential form

The functions are in the form $f(x) = a \cdot b^x$, where $b$ is the decay factor.

Step2: Determine the condition for rapid decrease

For exponential decay ($0 < b < 1$), the function decreases more rapidly as the base $b$ becomes smaller.

Step3: Compare the bases of the given functions

The bases are: a. $\frac{1}{2} = 0.5$ b. $\frac{3}{4} = 0.75$ c. $\frac{9}{10} = 0.9$ d. $\frac{5}{6} \approx 0.833$

Step4: Select the smallest base

Comparing the values: $0.5 < 0.75 < 0.833 < 0.9$. The smallest base is $0.5$.

Answer:

a. $f(x) = 2 \cdot \left(\frac{1}{2}\right)^x$