Question 3 If \( f(x)=x+4 \) and \( g(x)=x-4 \), (a) \( f(g(x))=\square \) (b) \( g(f(x))=\square \) (c) Thus \( g(x) \) is called an

To compute \( f(g(x)) \), we substitute \( g(x) \) into \( f \): \[ f(g(x)) = f(x-4) = (x-4) + 4 = x. \] For \( g(f(x)) \), we substitute \( f(x) \) into \( g \): \[ g(f(x)) = g(x+4) = (x+4) - 4 = x. \] Hence, \( g(x) \) is called an **inverse function** of \( f(x) \) because \( f(g(x)) \) and \( g(f(x)) \) both simplify to \( x \). When it comes to the relationship between a function and its inverse, it was actually conceptualized long before calculators were a thing. The idea comes from the need to reverse operations! It's like having a magic spell that turns addition into subtraction and vice versa. This has been crucial in algebra for solving equations. In practical scenarios such as computer programming, inverse functions can be particularly useful, especially in algorithms that require undo functionalities or reversing transformations. For instance, if a function encodes data, its inverse will decode it back, allowing you to retrieve the original data efficiently!