Let's analyze the composition $f(g(x))$ step by step:

1. Start with $g(x)$: $g(x) = 2x$ So, for any input $x$, the function $g(x)$ scales $x$ by 2.

2. Apply $f(x)$ to $g(x)$: The composition $f(g(x))$ means we take the result of $g(x)$ and plug it into $f(x)$: $f(g(x)) = f(2x)$ Since $f(x) = 4x$, substitute $2x$ into $f(x)$: $f(2x) = 4(2x) = 8x$ So, the composition $f(g(x)) = 8x$.

Effect on the Graph of $g(x)$:

• The graph of $g(x) = 2x$ is a straight line with a slope of 2, meaning it stretches the graph of $y = x$ by a factor of 2.
• The graph of $f(g(x)) = 8x$ further stretches this graph by a factor of 4 (from 2x to 8x).

In summary: The composition $f(g(x))$ has the effect of stretching the graph of $g(x) = 2x$ vertically by an additional factor of 4, resulting in an overall stretch by a factor of 8.

Would you like a detailed explanation on this or have any other questions?

Here are some related questions:

1. What is the effect of the composition $g(f(x))$?
2. How would the composition $f(f(x))$ affect the graph of $f(x)$?
3. What happens if we reverse the order of composition: $g(f(x))$?
4. How does the slope change affect the rate of change of these linear functions?
5. Can we generalize this to compositions of non-linear functions?

Tip: For linear functions, compositions often result in a simple multiplication of slopes.