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


Given the functions \( f(x) = 4x \) and \( g(x) = 2x \), let's analyze the composition \( f(g(x)) \) and determine its effect on the graph of \( g(x) \).

### Step 1: Find \( f(g(x)) \)
- \( g(x) = 2x \)
- Substitute \( g(x) \) into \( f(x) \):
  \[
  f(g(x)) = f(2x) = 4 \times (2x) = 8x
  \]

### Step 2: Analyze the effect on the graph of \( g(x) \)
- The original function \( g(x) = 2x \) has a slope of 2.
- The composition \( f(g(x)) = 8x \) has a slope of 8.

Comparing these slopes:
- The graph of \( g(x) \) is **stretched vertically** by a factor of 4 because the slope increases from 2 to 8.

Thus, the correct answer is:

**B. \( g(x) \) is stretched vertically.**

Would you like more details or have any other questions? Here are some related questions you might consider:

1. What happens to the graph of \( g(x) \) if you compose it with a function that divides by 2?
2. How do horizontal stretches and compressions differ from vertical ones?
3. What is the effect of adding a constant to \( f(x) \) before composing it with \( g(x) \)?
4. How would the composition change if \( f(x) \) were a quadratic function?
5. What is the general effect of composing a linear function with another linear function?

**Tip:** Always pay attention to how the composition of functions affects the slope, as it directly influences the vertical stretch or compression.

Understand how the composition of functions f(g(x)) affects the graph of g(x). Analyze the functions f(x) = 4x and g(x) = 2x to determine the vertical stretch in the graph of g(x) due to the composition. Suitable for students in Grades 9-12.

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