(e) Given the function \( f(x)=\log _{\frac{1}{4}} x \). (1) Sketch \( y=f(x) \) indicating an intercept with one of the axes and one othe point on the graph. (2) If \( f(x) \) is translated so that its new equation becomes \( h(x)=\log _{\frac{1}{4}}(x+3) \) state the domain of \( h(x) \). (3) State the equation of \( f^{-1} \) in the form \( g(x)=\ldots \)

Let's solve the problem step by step. ### Part (1): Sketch \( y = f(x) = \log_{\frac{1}{4}} x \) To sketch the graph of \( f(x) \), we need to understand the properties of the logarithmic function with a base less than 1. The function \( f(x) = \log_{\frac{1}{4}} x \) is defined for \( x > 0 \) and has the following characteristics: - **Intercept with the axes**: - The y-intercept occurs when \( x = 1 \): \[ f(1) = \log_{\frac{1}{4}} 1 = 0 \] - Therefore, the point \( (1, 0) \) is on the graph. - **Another point on the graph**: - Let's find \( f(4) \): \[ f(4) = \log_{\frac{1}{4}} 4 \] Since \( \frac{1}{4} = 4^{-1} \), we can use the change of base formula: \[ f(4) = -1 \] - Thus, the point \( (4, -1) \) is also on the graph. The graph of \( f(x) \) will decrease as \( x \) increases, approaching negative infinity as \( x \) approaches 0 from the right. ### Part (2): Domain of \( h(x) = \log_{\frac{1}{4}}(x + 3) \) The function \( h(x) \) is defined when the argument of the logarithm is positive: \[ x + 3 > 0 \] Solving this inequality: \[ x > -3 \] Thus, the domain of \( h(x) \) is: \[ (-3, \infty) \] ### Part (3): Equation of \( f^{-1} \) To find the inverse function \( f^{-1} \), we start with the equation: \[ y = \log_{\frac{1}{4}} x \] We can rewrite this in exponential form: \[ x = \left(\frac{1}{4}\right)^y \] Now, we can express \( y \) in terms of \( x \): \[ y = \log_{4} \left(\frac{1}{x}\right) \] This can be simplified to: \[ y = -\log_{4} x \] Thus, the inverse function is: \[ g(x) = -\log_{4} x \] ### Summary of Results 1. The graph of \( f(x) = \log_{\frac{1}{4}} x \) has intercepts at \( (1, 0) \) and \( (4, -1) \). 2. The domain of \( h(x) = \log_{\frac{1}{4}}(x + 3) \) is \( (-3, \infty) \). 3. The equation of the inverse function is \( g(x) = -\log_{4} x \).