1. **State the problem:** We have the function $g(x) = \log_2(4x)$ with $x > 0$. We need to find values of $g(4)$ and $g(\frac{1}{4})$, find the inverse function $g^{-1}(x)$, evaluate $g^{-1}(1)$, and describe the transformations for $y = g(x^2)$ from $y = \log_2 x$. 2. **Find $g(4)$:** $$g(4) = \log_2(4 \times 4) = \log_2(16)$$ Since $16 = 2^4$, $$g(4) = 4$$ 3. **Find $g(\frac{1}{4})$:** $$g\left(\frac{1}{4}\right) = \log_2\left(4 \times \frac{1}{4}\right) = \log_2(1)$$ Since $\log_2(1) = 0$, $$g\left(\frac{1}{4}\right) = 0$$ 4. **Find the inverse function $g^{-1}(x)$:** Start with $$y = \log_2(4x)$$ Rewrite in exponential form: $$2^y = 4x$$ Solve for $x$: $$x = \frac{2^y}{4}$$ Using cancellation, $$x = \frac{2^y}{\cancel{4}} = 2^{y-2}$$ So, $$g^{-1}(x) = 2^{x-2}$$ 5. **Find $g^{-1}(1)$:** Substitute $x=1$ into the inverse: $$g^{-1}(1) = 2^{1-2} = 2^{-1} = \frac{1}{2}$$ 6. **Describe transformations for $y = g(x^2)$ from $y = \log_2 x$:** - Start with $y = \log_2 x$. - Replace $x$ by $x^2$ to get $y = \log_2(x^2)$, which reflects a horizontal compression and symmetry about the y-axis. - Then apply $g(x) = \log_2(4x)$, which multiplies the input by 4 inside the log, shifting the graph horizontally. - So the combined transformation is: 1. Horizontal compression by squaring $x$. 2. Horizontal shift/compression by factor 4 inside the log. **Final answers:** (i) $g(4) = 4$ (ii) $g(\frac{1}{4}) = 0$ (b) $g^{-1}(x) = 2^{x-2}$ (c) $g^{-1}(1) = \frac{1}{2}$ (d) Transformations: replace $x$ by $x^2$ (horizontal compression and symmetry), then multiply input by 4 inside the log (horizontal scaling).