1. The problem asks to find the equation of the function $g(x)$ given the graph and the function $f(x) = -2 \log_2(x + 1) + 3$. 2. From the graph description, $g(x)$ is an increasing curve crossing near the origin and extending toward positive $y$-values as $x$ increases. 3. Since $f(x)$ is a transformed logarithmic function, and $g(x)$ is dashed and increasing, it is likely $g(x)$ is the inverse of $f(x)$. 4. To find the inverse, start with $y = f(x) = -2 \log_2(x + 1) + 3$. 5. Swap $x$ and $y$ to find $g(x)$: $$x = -2 \log_2(y + 1) + 3$$ 6. Solve for $y$: $$x - 3 = -2 \log_2(y + 1)$$ 7. Divide both sides by $-2$: $$\frac{x - 3}{-2} = \log_2(y + 1)$$ 8. Using the cancellation notation: $$\frac{\cancel{x - 3}}{\cancel{-2}} = \log_2(y + 1)$$ 9. Rewrite: $$\log_2(y + 1) = \frac{3 - x}{2}$$ 10. Convert from logarithmic to exponential form: $$y + 1 = 2^{\frac{3 - x}{2}}$$ 11. Finally, solve for $y$: $$y = 2^{\frac{3 - x}{2}} - 1$$ 12. Therefore, the equation of $g(x)$ is: $$g(x) = 2^{\frac{3 - x}{2}} - 1$$ This matches the behavior of the inverse of $f(x)$, an increasing function crossing near the origin.