onsider the function \( f(x)=\left(\frac{1}{3}\right)^{x} \). . Is \( f \) an increasing or decreasing function? Give a reason for your anower .2 Determine \( f^{-1}(x) \) in the form \( y=- \) .3 Write down the equation of the asymptote of \( f(x)-5 \). .4 Describe the transformation from \( f \) in \( g \) if \( g(x)=\log _{3} x \).

Let's solve the problem step by step. ### 1. Determine if \( f(x) \) is increasing or decreasing The function given is \( f(x) = \left(\frac{1}{3}\right)^{x} \). To determine if the function is increasing or decreasing, we can analyze the base of the exponential function. Since \( \frac{1}{3} < 1 \), the function is a decreasing exponential function. **Reason**: In general, if \( 0 < a < 1 \), then \( f(x) = a^x \) is a decreasing function because as \( x \) increases, \( a^x \) decreases. ### 2. Determine \( f^{-1}(x) \) To find the inverse function \( f^{-1}(x) \), we start with the equation: \[ y = \left(\frac{1}{3}\right)^{x} \] Now, we will solve for \( x \) in terms of \( y \): 1. Take the logarithm of both sides: \[ \log_{3}(y) = \log_{3}\left(\left(\frac{1}{3}\right)^{x}\right) \] 2. Using the property of logarithms, we can simplify: \[ \log_{3}(y) = x \cdot \log_{3}\left(\frac{1}{3}\right) \] 3. Since \( \log_{3}\left(\frac{1}{3}\right) = -1 \): \[ \log_{3}(y) = -x \] 4. Rearranging gives: \[ x = -\log_{3}(y) \] Thus, the inverse function is: \[ f^{-1}(x) = -\log_{3}(x) \] ### 3. Write down the equation of the asymptote of \( f(x) - 5 \) The function \( f(x) = \left(\frac{1}{3}\right)^{x} \) approaches \( 0 \) as \( x \) approaches infinity. Therefore, the horizontal asymptote of \( f(x) \) is \( y = 0 \). For the function \( f(x) - 5 \): \[ f(x) - 5 = \left(\frac{1}{3}\right)^{x} - 5 \] The horizontal asymptote will shift down by 5 units. Thus, the equation of the asymptote is: \[ y = -5 \] ### 4. Describe the transformation from \( f \) to \( g \) where \( g(x) = \log_{3}(x) \) The function \( f(x) = \left(\frac{1}{3}\right)^{x} \) is an exponential function, while \( g(x) = \log_{3}(x) \) is a logarithmic function. The transformation from \( f \) to \( g \) can be described as follows: - The function \( g(x) \) is the inverse of \( f(x) \). - Specifically, \( g(x) \) represents a reflection of \( f(x) \) across the line \( y = x \). In summary, the transformation from \( f \) to \( g \) is a reflection across the line \( y = x \). ### Final Summary 1. \( f(x) \) is a decreasing function. 2. \( f^{-1}(x) = -\log_{3}(x) \). 3. The asymptote of \( f(x) - 5 \) is \( y = -5 \). 4. The transformation from \( f \) to \( g \) is a reflection across the line \( y = x \).