Given the functions $f(x) = x^2$ and $g(x) = 4x$, we are tasked with finding $(f \circ g)(x)$, which represents the composition of the functions $f$ and $g$.

Step 1: Define the composition

The composition $(f \circ g)(x)$ means that we first apply $g(x)$ and then apply $f$ to the result. Symbolically, this can be written as: $(f \circ g)(x) = f(g(x))$

Step 2: Substitute $g(x)$ into $f(x)$

Since $g(x) = 4x$, we substitute $4x$ into the expression for $f(x) = x^2$: $(f \circ g)(x) = f(4x) = (4x)^2$

Step 3: Simplify the expression

Now, simplify $(4x)^2$: $(4x)^2 = 16x^2$

Final Answer:

$(f \circ g)(x) = 16x^2$

Would you like any further details or clarification on this? Here are five related questions for deeper exploration:

1. What is the domain of $(f \circ g)(x)$?
2. How would you find $(g \circ f)(x)$?
3. What happens if $f(x)$ is a different function, say $f(x) = x^3$?
4. How can you graph the composition of two functions like $f(x)$ and $g(x)$?
5. Can you generalize the process of composing two arbitrary functions?

Tip: Always remember to perform the function on the inside of the composition first and simplify step by step.