1. The problem asks to analyze the functions $f(x) = 2^x$ and $g(x) = 2^{-x}$ and understand their behavior and relationship. 2. Recall the exponential function properties: for any base $a > 0$, $a^{-x} = \frac{1}{a^x}$. 3. Applying this to $g(x)$, we have: $$g(x) = 2^{-x} = \frac{1}{2^x}$$ 4. This means $g(x)$ is the reciprocal of $f(x)$. 5. Since $f(x) = 2^x$ is an increasing exponential function (grows as $x$ increases), $g(x) = 2^{-x}$ is a decreasing exponential function (decreases as $x$ increases). 6. Both functions have the same value at $x=0$: $$f(0) = 2^0 = 1$$ $$g(0) = 2^{-0} = 1$$ 7. The graphs are symmetric with respect to the y-axis because $g(x) = f(-x)$. Final answer: $f(x) = 2^x$ is an increasing exponential function, and $g(x) = 2^{-x} = \frac{1}{2^x}$ is a decreasing exponential function, symmetric to $f(x)$ about the y-axis.