1. **Problem statement:** We want to find the value(s) of $k$ such that the equation $f(x) = k$ has exactly one solution. 2. **Understanding the problem:** The equation $f(x) = k$ represents a horizontal line intersecting the graph of $f(x)$. Having exactly one solution means the line touches the graph at exactly one point, i.e., the line is tangent to the graph. 3. **Key idea:** For $f(x) = k$ to have exactly one solution, the horizontal line $y = k$ must be tangent to the curve $y = f(x)$. 4. **Analyzing the options:** Since the problem does not provide the explicit function $f(x)$, we infer from the options that the values $k = -3, 1, 5, 9$ are candidates. 5. **Conclusion:** The value of $k$ for which $f(x) = k$ has exactly one solution is the value where the horizontal line touches the graph at exactly one point. Without the explicit function, we cannot determine which $k$ satisfies this, but the problem implies only one of these values is correct. **Final answer:** The value of $k$ could be **1** (option B) if the graph of $f(x)$ is tangent to the line $y=1$ at exactly one point. This is a typical scenario where the horizontal line intersects the graph once, indicating a single solution.