How do you find (g ∘ f)(x) for f(x) = 3x and g(x) = x^2?

To find \((g \circ f)(x)\), substitute \(f(x)\) into \(g(x)\), giving \((g \circ f)(x) = (3x)^2\).

In more detail, let's break down the process step by step. The notation \((g \circ f)(x)\) means we need to apply the function \(f(x)\) first and then apply the function \(g(x)\) to the result of \(f(x)\). Given the functions \(f(x) = 3x\) and \(g(x) = x^2\), we start by finding \(f(x)\).

First, calculate \(f(x)\):
\[ f(x) = 3x \]

Next, we need to substitute this result into the function \(g(x)\). This means wherever we see \(x\) in \(g(x)\), we replace it with \(f(x)\):
\[ g(f(x)) = g(3x) \]

Since \(g(x) = x^2\), we substitute \(3x\) for \(x\) in \(g(x)\):
\[ g(3x) = (3x)^2 \]

Now, simplify the expression:
\[ (3x)^2 = 9x^2 \]

Therefore, \((g \circ f)(x) = 9x^2\).

This process is called function composition, where one function is applied to the result of another function. It’s a useful technique in mathematics for combining functions and analysing their combined effect.