1. The problem involves understanding the transformations of the exponential function $f(x) = \left(\frac{1}{2}\right)^x$ and identifying the correct form of $g(x)$ after a translation. 2. The base function is $f(x) = \left(\frac{1}{2}\right)^x$. A horizontal translation of $h$ units to the left changes the function to $g(x) = \left(\frac{1}{2}\right)^{x + h}$. 3. Since the graph is translated 3 units to the left, $h = 3$, so the function becomes: $$g(x) = \left(\frac{1}{2}\right)^{x + 3}$$ 4. The other options represent different transformations: - $g(x) = \left(\frac{1}{2}\right)^{x - 3}$ translates 3 units to the right. - $g(x) = \left(\frac{1}{2}\right)^x - 3$ translates 3 units down. - $g(x) = \left(\frac{1}{2}\right)^x + 3$ translates 3 units up. 5. Therefore, the correct function after translating $f(x)$ three units to the left is: $$g(x) = \left(\frac{1}{2}\right)^{x + 3}$$