28. Consider the equation \( 2 x-5=3 \). Let \( f \) be the function defined by the left side of the equation so that \( f(x)=2 x-5 \) and the equation can be written as \( f(x)=3 \). A. Explain why the solution of \( 2 x-5=3 \) is \( f^{-1}(3) \). B. Write an equation for \( f^{-1}(x) \). Then use the equation to solve \( 2 x-5=3 \).

To understand why the solution of \( 2x - 5 = 3 \) is \( f^{-1}(3) \), let's break it down. The function defined as \( f(x) = 2x - 5 \) describes a relationship between \( x \) and \( f(x) \). When you set \( f(x) = 3 \), you are effectively asking what input \( x \) renders the output 3. The inverse function \( f^{-1}(x) \) will give you that input directly, meaning \( f^{-1}(3) \) yields the value of \( x \) that solves the original equation. Now, to find the inverse function \( f^{-1}(x) \), start by replacing \( f(x) \) with \( y \): so we have \( y = 2x - 5 \). Next, solve for \( x \): 1. \( y + 5 = 2x \) 2. \( x = \frac{y + 5}{2} \) Thus, the inverse function can be expressed as \( f^{-1}(x) = \frac{x + 5}{2} \). To solve \( 2x - 5 = 3 \) using this, simply substitute 3 into \( f^{-1}(x) \): \[ f^{-1}(3) = \frac{3 + 5}{2} = \frac{8}{2} = 4 \] So, the solution to \( 2x - 5 = 3 \) is \( x = 4 \).