If f(x)=4x
 and g(x)=2x
, what is the effect of the composition f(g(x))
 on the graph of g(x)
?
A.g(x) is compressed vertically.
B.g(x) is stretched vertically.
C.g(x) is translated 4 units up.
D.g(x) is translated 4 units to the left.

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

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

The composition \( f(g(x)) \) means plugging \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f(2x) = 4(2x) = 8x
\]

Now, let's analyze the transformation:

- The function \( g(x) = 2x \) has a slope of 2, meaning it stretches the graph vertically by a factor of 2.
- The composition \( f(g(x)) = 8x \) has a slope of 8, which means it further stretches the graph vertically by a factor of 8 compared to \( g(x) \).

Since the slope of \( f(g(x)) \) is greater than the slope of \( g(x) \), this indicates a **vertical stretch** of \( g(x) \).

Therefore, the correct answer is:
**B. g(x) is stretched vertically.**

Would you like a deeper explanation or have any more questions?

Here are 5 related questions to explore:
1. What is the effect of composing two linear functions on their slopes?
2. How does vertical stretching differ from horizontal stretching?
3. What happens if you compose two quadratic functions instead of linear functions?
4. What is the impact of negative coefficients on vertical transformations?
5. How can you determine transformations using function compositions in general?

**Tip:** In compositions, the order of functions matters—applying transformations step by step can help visualize the combined effect.

If f(x)=4x and g(x)=2x, what is the effect of the composition f(g(x)) on the graph of g(x)? A. g(x) is compressed vertically. B. g(x) is stretched vertically. C. g(x) is translated 4 units up. D. g(x) is translated 4 units to the left.

Understand the effect of composing linear functions f(x) = 4x and g(x) = 2x, and how it stretches g(x) vertically. This detailed explanation covers function composition, graph transformations, and the impact of slopes. Suitable for students in Grades 9-12.

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