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

To analyze the effect of the composition \( f(g(x)) \) on the graph of \( g(x) \), let's first determine what the composition \( f(g(x)) \) looks like:

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

The composition \( f(g(x)) \) is found by substituting \( g(x) \) into \( f(x) \):

\[
f(g(x)) = f(2^x) = 4 \cdot 2^x
\]

This means that \( f(g(x)) = 4 \cdot 2^x \).

### Effect on the Graph of \( g(x) \):
- The graph of \( g(x) = 2^x \) is an exponential function, which increases rapidly as \( x \) increases.
- The composition \( f(g(x)) = 4 \cdot 2^x \) is also an exponential function but with a multiplicative factor of 4.

**Effect:**
- The composition \( f(g(x)) \) results in a vertical stretch of the graph of \( g(x) \) by a factor of 4. 
- This means that every point on the graph of \( g(x) = 2^x \) will be multiplied by 4 in the \( y \)-direction.

In summary, the graph of \( f(g(x)) = 4 \cdot 2^x \) is the same as the graph of \( g(x) = 2^x \), but it is stretched vertically by a factor of 4.

Would you like more details or have any questions?

### Relative Questions:
1. How would the composition \( g(f(x)) \) affect the graph of \( f(x) \)?
2. What would happen to the graph of \( g(x) = 2^x \) if it was multiplied by a different constant, say 10?
3. How does the graph of \( g(x) = 2^x \) compare to the graph of \( h(x) = 3^x \)?
4. What is the effect of applying a horizontal shift to the graph of \( g(x) = 2^x \)?
5. Can you describe how the graph of \( f(x) = 4x \) would look compared to \( g(x) = 2^x \)?

### Tip:
When dealing with compositions of functions, analyze the inner function first and then apply the outer function to understand the transformation on the graph.

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

Understand the effect of the composition of functions f(x) = 4x and g(x) = 2^x on the graph of g(x). Learn how to analyze and describe transformations, including vertical stretching of exponential functions. Suitable for high school students in Grades 9-12.

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