Find the inverse function for \( f(x)=\frac{8}{2 x-1} \) There is no inverse \( f^{-1}(x)=\frac{2 x-1}{8} \) \( f^{-1}(x)=\frac{4}{x}+\frac{1}{2} \) \( f^{-1}(x)=\frac{4}{x}-\frac{1}{2} \)

To find the inverse function of \( f(x)=\frac{8}{2x-1} \), let’s set \( y = f(x) \). Rewriting it gives us \( y = \frac{8}{2x - 1} \). To find the inverse, we swap \( x \) and \( y \) and solve for \( y \): 1. Start with \( x = \frac{8}{2y - 1} \). 2. Rearranging gives \( x(2y - 1) = 8 \) or \( 2xy - x = 8 \). 3. Thus, \( 2xy = x + 8 \) leading to \( y = \frac{x + 8}{2x} \). So the inverse function is \( f^{-1}(x) = \frac{8}{2x} + \frac{1}{2} \), which simplifies to \( f^{-1}(x)=\frac{4}{x}-\frac{1}{2} \). Knowing how to find inverse functions is super handy! For instance, if you were dealing with real-world scenarios like electric circuits or plumbing problems, understanding the inversion might help backtrack to the original situation based on results. If you’re just getting comfortable with inverses, be cautious with mistakes. A common one is neglecting the domain. When you find an inverse, check if the original and the inverse can be expressed in terms of the same x-values; it ensures that you’re truly reversing the function without losing information!