Are the following functions inverses? \[ \begin{array}{l} f(x)=2 x-1 \\ g(x)=\frac{x+1}{2} \end{array} \] No, they are not inverses Yes, they are inverses

To determine if \( f(x) \) and \( g(x) \) are inverses, we can check if \( f(g(x)) = x \) and \( g(f(x)) = x \). First, calculate \( f(g(x)) \): \[ f(g(x)) = f\left(\frac{x+1}{2}\right) = 2\left(\frac{x+1}{2}\right) - 1 = x + 1 - 1 = x. \] Now, calculate \( g(f(x)) \): \[ g(f(x)) = g(2x - 1) = \frac{(2x - 1) + 1}{2} = \frac{2x}{2} = x. \] Since both compositions equal \( x \), we can confirm that yes, they are indeed inverses! Both functions beautifully embody the relationship where one undoes the work of the other. Think of this partnership like a perfect dance duo where each step complements the other, flawlessly returning to the starting point!