To solve this problem, we are tasked with determining the composite function $y = f(g(x))$, where $f$ is given as a graph and $g(x) = 2x$.

Step-by-Step Solution:

1. Understand $g(x) = 2x$:

   • This function scales every input $x$ by 2. So for $x$, the value $g(x) = 2x$ will "stretch" the inputs.

2. Composite Function $y = f(g(x))$:

   • To evaluate $f(g(x))$, we plug $g(x) = 2x$ into the function $f$.
   • This means instead of looking at $f(x)$, we look at $f(2x)$. The $x$-coordinates on the graph of $f$ are "compressed" (by a factor of 2).

3. Effect of $g(x) = 2x$ on $f$:

   • The graph of $f$, defined on the $x$-axis, will now be evaluated at half the $x$-values.
   • For instance:
     • $f(2x)$ at $x = 1$ is the same as $f(2 \cdot 1) = f(2)$.
     • $f(2x)$ at $x = 2$ corresponds to $f(4)$.

4. Re-draw the graph $f(g(x))$:

   • The key $x$-values of $f$ need to be "compressed" to half their original values on the $x$-axis.
   • For instance:
     • $f(0)$ stays the same.
     • $f(1)$ (originally at $x = 2$) will now occur at $x = 1$.
     • $f(2)$ (originally at $x = 4$) will now occur at $x = 2$.

Compare to the Given Choices:

• By looking closely at the provided options, Option C represents the correct graph:
  • The $x$-values are compressed by a factor of 2.
  • The $y$-values remain the same.

Final Answer: Option C

Would you like me to further explain how the scaling works, or go into more detail about function composition? Let me know!

Related Questions:

1. What happens when a function is composed with a horizontal stretch instead of a compression?
2. How does the graph of $y = f(x)$ change when replaced with $y = f(2x)$?
3. What is the effect of scaling a function vertically versus horizontally?
4. How do you identify composite functions from a graph?
5. What is the difference between $f(g(x))$ and $g(f(x))$?

Tip: For $y = f(ax)$, the graph compresses horizontally by $a$ when $a > 1$ and stretches when $0 < a < 1$.