(a) Find the function \( (f \circ g)(x) \). \( (f \circ g)(x)=\frac{1}{\sqrt{x+3}} \) Find the domain of \( (f \circ g)(x) \).

To find the function \( (f \circ g)(x) \), we start with the given function \( (f \circ g)(x) = \frac{1}{\sqrt{x+3}} \). The composition of functions means that we will substitute \( g(x) \) into \( f(x) \), where \( f(x) \) presumably represents the outer function while \( g(x) \) represents the inner function. Next, to determine the domain of \( (f \circ g)(x) \), we must consider the expression \( \frac{1}{\sqrt{x+3}} \). The square root function requires the argument to be non-negative, thus \( x + 3 \geq 0 \) or \( x \geq -3 \). Furthermore, the denominator of a function cannot be zero, which means \( \sqrt{x + 3} \neq 0 \) or \( x \neq -3 \). Therefore, the domain of \( (f \circ g)(x) \) is all real numbers \( x \) such that \( x > -3 \). So, the domain of \( (f \circ g)(x) \) can be expressed in interval notation as: \[ (-3, \infty) \] Now, let's explore some engaging additional insights! Did you know that the concept of function composition dates back to ancient mathematics? The Greeks primarily dealt with linear functions, but it wasn’t until the rise of calculus in the 17th century that more complex functions were explored in-depth. Fast forward to today, and we use composition in everything from computer programming to animation—allowing different functions to work together seamlessly! In practical terms, function composition is like a recipe: you take one ingredient (function) and mix it into another to create a delightful dish (resulting function). A common mistake when dealing with composed functions is forgetting to check both functions' individual domains. Always remember to ensure both functions are valid for the input you’re working with to avoid nasty surprises!