The diagram shows the graph of the function $f(x) = ext{log}_3 x$, where $x > 0$ - Scottish Highers Maths - Question 9 - 2023

To sketch the graph of the inverse function $f^{-1}(x)$, we first need to reflect the graph of the original function $f(x)$ across the line $y = x$. The function $f(x) = ext{log}_3 x$ passes through the points (1, 0) and (3, 1), so reflecting these points gives us (0, 1) and (1, 3) respectively.

Next, we apply the translation downwards by 1 unit to obtain the graph of $y = f^{-1}(x) - 1$. This translates the reflected points we calculated earlier. Therefore, the point (0, 1) becomes (0, 0) and the point (1, 3) becomes (1, 2).

The resulting graph will approach the line $y = x - 1$ asymptotically as $x$ approaches infinity. The sketch should clearly demonstrate a concave up curve that passes through (0, 0) and (1, 2). This concave shape indicates that the function is increasing, which aligns with the properties of inverse functions.