21. (B) \( f(x)=-2 x+6 \), then \( f^{-1}(x)= \) : \( \begin{array}{llll}\text { (A) } 6-2 x & \text { (B) } \frac{6-x}{2} & \text { (C) }-2 & \text { (D) } 1\end{array} \) (C) \( \frac{2}{6-x} \)

To find the inverse of the function \( f(x) = -2x + 6 \), we start by replacing \( f(x) \) with \( y \): \[ y = -2x + 6. \] Next, we swap \( x \) and \( y \): \[ x = -2y + 6. \] Now, solving for \( y \): 1. Rearranging the equation gives us \( -2y = x - 6 \). 2. Dividing both sides by -2, we get \( y = \frac{6 - x}{2} \). Thus, the inverse function is: \[ f^{-1}(x) = \frac{6 - x}{2}. \] In this case, option (B) is correct! Conversely, the concept of function inverses has historical roots in the study of algebra and calculus. Mathematicians like Descartes and Newton helped lay the groundwork for functions and their inverses. They explored how to “undo” operations, a fundamental part of modern mathematics. For real-world application, understand that the concept of function inverses is prevalent in fields like economics, where supply and demand can be modeled. When you have a demand function, finding its inverse can help you determine what price will lead to a certain quantity being demanded, making it valuable for pricing strategies.