44. What is the inverse of \( f(x)=2^{x}-3 \) ? A. \( f^{-1}(x)=\log _{3}(x+2) \) B. \( f^{-1}(x)=\log _{2}(x+3) \) C. \( f^{-1}(x)=\log _{3}(x-2) \) D. \( f^{-1}(x)=\log _{2}(x-3) \)

To find the inverse of the function \( f(x) = 2^{x} - 3 \), we start by replacing \( f(x) \) with \( y \): 1. Set \( y = 2^x - 3 \). 2. To find the inverse, solve for \( x \): - \( y + 3 = 2^x \) - Take the logarithm base 2: \( x = \log_2(y + 3) \). Thus, the inverse function is \( f^{-1}(x) = \log_2(x + 3) \), which corresponds to option B! This exercise highlights the importance of logarithms in reversing exponential functions, a key concept in mathematics often encountered in various scientific disciplines! Additionally, the concept of inverse functions is not just limited to mathematics. In real-world applications, understanding how to reverse processes can help us in fields like engineering, where one may need to predict the input values required to achieve desired results based on output conditions—think of it as a balance between cause and effect!