1. Sketch the graph of \( y=4^{x} \) and the inverse on the same system of axis Given: \( f(x)=5^{\mathrm{x}} \) a) Write down the equation of the inverse of \( f(x) \). b) Sketch \( f \) and \( f^{-1} \) on the same system of axes. c) Write down the equation of the horizontal asymptote of \( f \). d) Write down the equation of the vertical asymptote of \( f-1 \). e) Give the domain \( f \). f) Give the domain of \( f^{-1} \).

Let's solve the problem step by step. ### Step 1: Sketch the graph of \( y = 4^x \) and its inverse The function \( y = 4^x \) is an exponential function that increases rapidly as \( x \) increases. Its inverse can be found by switching \( x \) and \( y \) and solving for \( y \). ### Step 2: Find the inverse of \( f(x) = 5^x \) To find the inverse, we set \( y = 5^x \) and switch \( x \) and \( y \): 1. Start with \( x = 5^y \). 2. Take the logarithm of both sides: \( \log_5(x) = y \). Thus, the inverse function is: \[ f^{-1}(x) = \log_5(x) \] ### Step 3: Sketch \( f \) and \( f^{-1} \) The graph of \( f(x) = 5^x \) is an increasing curve that approaches the x-axis (horizontal asymptote) as \( x \) approaches negative infinity. The graph of \( f^{-1}(x) = \log_5(x) \) is an increasing curve that approaches the y-axis (vertical asymptote) as \( x \) approaches zero. ### Step 4: Write down the equations of the asymptotes **c)** The horizontal asymptote of \( f(x) = 5^x \) is: \[ y = 0 \] **d)** The vertical asymptote of \( f^{-1}(x) = \log_5(x) \) is: \[ x = 0 \] ### Step 5: Determine the domains **e)** The domain of \( f(x) = 5^x \) is all real numbers: \[ \text{Domain of } f: (-\infty, \infty) \] **f)** The domain of \( f^{-1}(x) = \log_5(x) \) is \( x > 0 \): \[ \text{Domain of } f^{-1}: (0, \infty) \] ### Summary of Results - **Inverse of \( f(x) \)**: \( f^{-1}(x) = \log_5(x) \) - **Horizontal Asymptote of \( f \)**: \( y = 0 \) - **Vertical Asymptote of \( f^{-1} \)**: \( x = 0 \) - **Domain of \( f \)**: \( (-\infty, \infty) \) - **Domain of \( f^{-1} \)**: \( (0, \infty) \) ### Graph Sketch To sketch the graphs of \( f(x) = 5^x \) and \( f^{-1}(x) = \log_5(x) \): 1. **Graph of \( f(x) = 5^x \)**: - Passes through (0, 1) since \( 5^0 = 1 \). - Approaches the x-axis as \( x \) goes to negative infinity. 2. **Graph of \( f^{-1}(x) = \log_5(x) \)**: - Passes through (1, 0) since \( \log_5(1) = 0 \). - Approaches the y-axis as \( x \) goes to 0. Both graphs are reflections of each other across the line \( y = x \). If you need a visual representation, please let me know!