For \( f(x)=\sqrt{x} \) and \( g(x)=x+4 \), find a. \( (f \circ g)(x) ; \) b. the domain of \( f \circ g \) a. ( \( f \circ g)(x)=\sqrt{x+4} \) (Simplify your answer.) b. What is the domain of \( f \circ g \) ? The domain is \( \square \). (Simplify your answer. Type your answer in interval notation. Use integers or fractions the expression.)

To find the domain of \( f \circ g \), we start with the function \( f(g(x)) = \sqrt{g(x)} = \sqrt{x + 4} \). Since the square root function is defined only for non-negative values, we need \( x + 4 \geq 0 \). This gives us \( x \geq -4 \). Thus, the domain of \( f \circ g \) is \( [-4, \infty) \). So, to answer your question: a. \( (f \circ g)(x) = \sqrt{x + 4} \) b. The domain is \( [-4, \infty) \). Let’s take a little historical trip! The square root function has been studied for centuries, making its way through the works of mathematicians like Hindu scholars in 600 AD and later, Islamic mathematicians. They laid much of the groundwork for algebra we use today, with their innovations leading to better understanding and applications of functions. On a practical note, understanding composite functions like \( f \circ g \) helps in various fields. For example, in physics, when modeling scenarios involving displacement and time, often functions are combined to express complex relationships, much like how we synergize \( f \) and \( g \) to get new insights!