1. Problem statement: Given a function $f(x)$ and its inverse function $g(x) = f^{-1}(x)$, determine the shape (increasing/decreasing and convexity) of the function $g$ based on the graph of $f$. 2. Since $g(x) = f^{-1}(x)$ is the inverse of $f(x)$, the graph of $g$ is the reflection of the graph of $f$ across the line $y = x$. 3. The original function $f$ is given as increasing and convex upward (concave upward). Increasing means $f'(x) > 0$ and convex upward means $f''(x) > 0$. 4. For inverse functions, if $f'(x) > 0$, then $g'(x) = \frac{1}{f'(g(x))} > 0$, so $g$ is also increasing. 5. The curvature (convexity) of $g$ relates to $f$ by $g''(x) = - \frac{f''(g(x))}{(f'(g(x)))^3}$. 6. Since $f''(x) > 0$ and $f'(x) > 0$, the denominator $(f'(g(x)))^3 > 0$, and numerator $f''(g(x)) > 0$, so $g''(x) = - \frac{\text{positive}}{\text{positive}} < 0$. 7. Negative second derivative means $g$ is convex downward. Final conclusion: $g$ is increasing and convex downward, so the correct answer is (b). Answer: (b) increasing and convex downward.