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

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.

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

Learn how the composition of two linear functions, f(x) = 4x and g(x) = 2x, affects the graph of g(x). This problem explores the concepts of function composition and linear transformations, suitable for students in Grades 9-10.

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