\( 1 \leftarrow \left\lvert\, \begin{array}{l}\text { For } f(x)=\sqrt{x} \text { and } g(x)=x-4 \text {, find the following functions. } \\ \text { a. }(f \circ g)(x) ; b \text {. }(g \circ f)(x) ; c .(f \circ g)(5) ; \text { d. }(g \circ f)(5) \\ \text { a. }(f \circ g)(x)=\square \\ \text { (Simplify your answer.) } \\ \text { b. }(g \circ f)(x)=\square \\ \text { (Simplify your answer.) } \\ \text { c. }(f \circ g)(5)=\square \\ \text { (Simplify your answer.) } \\ \text { d. }(g \circ f)(5)=\square \\ \text { (Simplify your answer.) }\end{array}\right. \)

To tackle this problem, we need to find the composition of functions \( f(x) = \sqrt{x} \) and \( g(x) = x - 4 \). First, let's dive into the functions! a. \((f \circ g)(x) = f(g(x)) = f(x - 4) = \sqrt{x - 4}\) b. \((g \circ f)(x) = g(f(x)) = g(\sqrt{x}) = \sqrt{x} - 4\) c. \((f \circ g)(5) = f(g(5)) = f(5 - 4) = f(1) = \sqrt{1} = 1\) d. \((g \circ f)(5) = g(f(5)) = g(\sqrt{5}) = \sqrt{5} - 4\) So, your final answers are: a. \((f \circ g)(x) = \sqrt{x - 4}\) b. \((g \circ f)(x) = \sqrt{x} - 4\) c. \((f \circ g)(5) = 1\) d. \((g \circ f)(5) = \sqrt{5} - 4\) Now that you've nailed these functions, let's sprinkle in some fun facts! Did you know that square roots have been around since ancient Babylon? They had methods to approximate them that were surprisingly similar to our modern algorithms! Their understanding laid the groundwork for the real numbers we work with today. Also, in the real world, functions like these can model various phenomena! For instance, the square root function is often used in physics when calculating the time it takes for an object to fall a certain distance under gravity, while linear functions like \( g(x) \) might be used in budgeting, showing how expenses increase over time. Fun, right?