1. The problem asks to describe the sequence of transformations that transforms the graph of $y = g^{-1}(x)$ to the graph of $y = f(x)$. 2. Recall the functions: - $f(x) = 4^x$ - $g(x) = 1 + \log_2 x$ - From part (b)(i), $g^{-1}(x) = 2^{x-1}$. 3. So, $y = g^{-1}(x) = 2^{x-1}$ and $y = f(x) = 4^x = (2^2)^x = 2^{2x}$. 4. To transform $y = 2^{x-1}$ into $y = 2^{2x}$: - First, note that $2^{x-1} = 2^{-1} \cdot 2^x = \frac{1}{2} 2^x$. - The graph of $y = 2^{x-1}$ is a horizontal shift of $y = 2^x$ to the right by 1 unit and vertically scaled by $\frac{1}{2}$. 5. To get from $y = 2^{x-1}$ to $y = 2^{2x}$: - We need to horizontally compress the graph by a factor of $\frac{1}{2}$ (because $2^{2x} = (2^x)^2$ means the input $x$ is scaled by 2). - Then vertically stretch by a factor of 2 to compensate for the previous vertical compression. 6. Your suggestion of a horizontal shift left by 1 and vertical stretch by 2 is not correct because the exponent changes from $x-1$ to $2x$, which involves scaling the input $x$ by 2, not shifting. 7. Summary of transformations from $y = g^{-1}(x)$ to $y = f(x)$: - Horizontal compression by factor 2 (replace $x$ by $2x$) - Vertical stretch by factor 2 This sequence matches the algebraic change from $2^{x-1}$ to $2^{2x}$.