Question

ponential decay functionsstandard (15261)two exponential functions are shown in the table.| $x$ | $f(x)=2^x$ | $g(x)=\left(\frac{1}{2}\
ight)^x$ ||-----|------------|------------------------------------|| 2 | 4 | $\frac{1}{4}$ || 1 | 2 | $\frac{1}{2}$ || 0 | 1 | 1 || -1 | $\frac{1}{2}$ | 2 || -2 | $\frac{1}{4}$ | 4 |which conclusion about $f(x)$ and $g(x)$ can be drawn from the table?- the function $f(x)$ is a decreasing function, and $g(x)$ is an increasing function.- the functions $f(x)$ and $g(x)$ are reflections over the $y$-axis.- the functions $f(x)$ and $g(x)$ are reflections over the $x$-axis.- the function $f(x)$ has a greater initial value than $g(x)$

Explanation:

1. Analyze $f(x)=2^x$: As $x$ increases (from -2 to 2), $f(x)$ increases from $\frac{1}{4}$ to 4, so it is an increasing function.
2. Analyze $g(x)=(\frac{1}{2})^x$: As $x$ increases (from -2 to 2), $g(x)$ decreases from 4 to $\frac{1}{4}$, so it is a decreasing function.
3. Check reflections: $g(x)=(\frac{1}{2})^x=2^{-x}$, which is a reflection of $f(x)=2^x$ over the y-axis, not the x-axis.
4. Check initial values: At $x=0$, $f(0)=1$ and $g(0)=1$, so their initial values are equal.

Answer:

The function $f(x)$ is a decreasing function, and $g(x)$ is an increasing function.