How do you find the composite function (f ∘ g)(x) for f(x) = x + 3 and g(x) = 2x?

To find the composite function (f ∘ g)(x), substitute g(x) into f(x), resulting in (f ∘ g)(x) = 2x + 3.

In more detail, a composite function is created when one function is applied to the result of another function. Here, we have two functions: \( f(x) = x + 3 \) and \( g(x) = 2x \). The notation \( (f ∘ g)(x) \) means that we first apply \( g(x) \) and then apply \( f(x) \) to the result of \( g(x) \).

To find \( (f ∘ g)(x) \), we start by substituting \( g(x) \) into \( f(x) \). Since \( g(x) = 2x \), we replace the \( x \) in \( f(x) \) with \( 2x \). This gives us:
\[ f(g(x)) = f(2x) \]

Next, we use the definition of \( f(x) \), which is \( x + 3 \). So, wherever we see \( x \) in \( f(x) \), we replace it with \( 2x \):
\[ f(2x) = 2x + 3 \]

Therefore, the composite function \( (f ∘ g)(x) \) is:
\[ (f ∘ g)(x) = 2x + 3 \]

This means that if you first apply \( g(x) \) to \( x \), doubling it to get \( 2x \), and then apply \( f(x) \) to \( 2x \), you add 3 to get the final result. This process of substitution is key to understanding how composite functions work.