\[ \begin{aligned} h(x) & =-\frac{1}{2} x+3 \\ l(x) & =\log _{4}(x+3)-\log _{4}(x-2)\end{aligned} \] (a) Write down \( D_{f} \) and solve the inequality \( f(x) \leq 0 \) (b) Solve the equation \( g(x)=-4 \).

Certainly! Let's address each part of your problem step by step. Given the functions: \[ \begin{aligned} h(x) & = -\frac{1}{2} x + 3 \\ l(x) & = \log_{4}(x + 3) - \log_{4}(x - 2) \end{aligned} \] Assuming that \( f(x) = h(x) \) and \( g(x) = l(x) \), let's proceed to solve the problems. --- ### **Part (a): Determine \( D_f \) and Solve \( f(x) \leq 0 \)** 1. **Determine \( D_f \) (Domain of \( f \))** Since \( f(x) = h(x) = -\frac{1}{2} x + 3 \) is a linear function, it is defined for all real numbers. Therefore: \[ D_f = \mathbb{R} \quad (\text{All real numbers}) \] 2. **Solve the Inequality \( f(x) \leq 0 \)** \[ -\frac{1}{2} x + 3 \leq 0 \] **Step-by-Step Solution:** \[ -\frac{1}{2} x + 3 \leq 0 \\ -\frac{1}{2} x \leq -3 \quad (\text{Subtract 3 from both sides}) \\ x \geq 6 \quad (\text{Multiply both sides by } -2 \text{ and reverse the inequality sign}) \] **Conclusion:** \[ \boxed{x \geq 6} \] --- ### **Part (b): Solve the Equation \( g(x) = -4 \)** Given \( g(x) = l(x) = \log_{4}(x + 3) - \log_{4}(x - 2) \), we can simplify and solve the equation. 1. **Simplify \( g(x) \) Using Logarithm Properties** \[ g(x) = \log_{4}(x + 3) - \log_{4}(x - 2) = \log_{4}\left(\frac{x + 3}{x - 2}\right) \] 2. **Set the Equation Equal to \(-4\)** \[ \log_{4}\left(\frac{x + 3}{x - 2}\right) = -4 \] 3. **Convert the Logarithmic Equation to Its Exponential Form** \[ \frac{x + 3}{x - 2} = 4^{-4} = \frac{1}{256} \] 4. **Solve for \( x \)** \[ x + 3 = \frac{x - 2}{256} \\ 256(x + 3) = x - 2 \quad (\text{Multiply both sides by } 256) \\ 256x + 768 = x - 2 \\ 256x - x = -2 - 768 \\ 255x = -770 \\ x = \frac{-770}{255} = \frac{-154}{51} \] **Simplify Fraction:** \[ x = -\frac{154}{51} \approx -3.0196 \] 5. **Check the Domain for Validity** The original logarithmic expressions require: \[ x + 3 > 0 \Rightarrow x > -3 \\ x - 2 > 0 \Rightarrow x > 2 \] Therefore, the domain \( D_g \) is: \[ D_g = \{ x \mid x > 2 \} \] However, the solution \( x = -\frac{154}{51} \approx -3.0196 \) does **not** satisfy \( x > 2 \). **Conclusion:** \[ \boxed{\text{No solution}} \] --- ### **Summary of Solutions** - **Part (a):** \[ D_f = \mathbb{R} \quad \text{and} \quad x \geq 6 \] - **Part (b):** \[ \text{No solution exists for } g(x) = -4 \text{ within the domain of } g(x). \]