If f(x)=4x and g(x)=2x, what is the effect of the composition f(g(x)) on the graph of g(x)?

To find the effect of the composition \( f(g(x)) \) on the graph of \( g(x) \), let's first compute \( f(g(x)) \).

Given:
- \( f(x) = 4x \)
- \( g(x) = 2x \)

Now, \( f(g(x)) \) means applying \( f(x) \) to the output of \( g(x) \), so:

\[
f(g(x)) = f(2x) = 4(2x) = 8x
\]

### Analysis:
- The graph of \( g(x) = 2x \) is a straight line with a slope of 2, meaning it increases at a rate of 2 units in the y-direction for every 1 unit in the x-direction.
- The composition \( f(g(x)) = 8x \) is also a straight line, but its slope is 8.

### Effect on the graph:
The composition of \( f(g(x)) \) scales the slope of \( g(x) \) by a factor of 4 (from 2 to 8). This results in a **steeper line**, indicating a more rapid increase in y-values for every x-value. The line becomes much steeper than the original \( g(x) \), and the graph stretches vertically by a factor of 4.

Would you like more details on this transformation or further examples?

Here are 5 related questions to extend your understanding:
1. What happens if \( f(x) = 3x \) and \( g(x) = x + 1 \)? What would \( f(g(x)) \) be?
2. How does horizontal translation affect the graph of the composition function?
3. What is the inverse of the composition \( f(g(x)) \)?
4. What if \( f(x) = x^2 \) and \( g(x) = 2x \)? How would the composition affect the graph?
5. Can the composition \( g(f(x)) \) yield a different result than \( f(g(x)) \)?

**Tip:** Always check if a composition of functions changes the slope, shifts the graph, or modifies its shape in some other way!

Explore the effect of the composition f(g(x)) on the graph of g(x), where f(x) = 4x and g(x) = 2x. Understand how the slope changes from 2 to 8, resulting in a steeper graph. Suitable for Grades 9-11 students learning about function composition and linear functions.

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