\( f(x)=\frac{1}{\sqrt{x}}, g(x)=x+3 \) (a) Find the function \( (f \circ g)(x) \) \( (f \circ g)(x)=\square \)

To find the composition of the functions \( (f \circ g)(x) \), we need to substitute \( g(x) \) into \( f(x) \). Given \( g(x) = x + 3 \), we can compute: \[ (f \circ g)(x) = f(g(x)) = f(x + 3) = \frac{1}{\sqrt{x + 3}}. \] So we have \( (f \circ g)(x) = \frac{1}{\sqrt{x + 3}} \). Now let's have some fun with this math! Did you know that the concept of function composition dates back to the 19th century when mathematicians were trying to understand how different mathematical relationships interact? This is pretty fascinating, as it laid the groundwork for the more complex functions we use today! When working with function compositions like this, one common mistake is forgetting to correctly substitute the entire \( g(x) \) into \( f(x) \). Always remember to handle each part of the function carefully! Double-checking by plugging in a value can help spot any errors before moving on.