Question:

Step 1: Understand what we're looking for. We need to find the number of solutions to the equation $f(x) = f^{-1}(x)$. This means we're looking for points where the function value equals the inverse function value. Step 2: Identify the key property of solutions. Solutions to $f(x) = f^{-1}(x)$ fall into two categories: - Points where $f(x) = x$ (these always satisfy the equation since $f^{-1}(x) = x$ at these points) - Points where $f(a) = b$ and $f(b) = a$ with $a \neq b$ (symmetric pairs about the line $y = x$) Step 3: Analyze intersections with $y = x$. From the given graph of $f(x)$ defined on $[-2, 2]$, we examine where the curve $y = f(x)$ intersects the line $y = x$. Each intersection point $(c, c)$ satisfies $f(c) = c$, which automatically satisfies $f(c) = f^{-1}(c)$. Step 4: Consider symmetric solutions about $y = x$. For points not on the line $y = x$, if $(a, b)$ is on the graph of $f$ (meaning $f(a) = b$), then $(b, a)$ is on the graph of $f^{-1}$. The equation $f(x) = f^{-1}(x)$ is also satisfied when the graph of $f$ intersects the graph of $f^{-1}$, which occurs on the line $y = x$ and potentially on other symmetric configurations. Step 5: Count solutions from the graph. Examining the provided figure carefully, the curve $y = f(x)$ intersects the line $y = x$ at exactly **3 points**. These are all the solutions to $f(x) = f^{-1}(x)$ for this particular function. **Final Answer:** The number of solutions to $f(x) = f^{-1}(x)$ is **3**. The answer is **Option 2: 3**