Let's analyze the problem you uploaded.

You are asked to match two functions, $f(x) = |\sin(x)|$ and $f(x) = \sin(|x|)$, with their corresponding graphs. Here's the breakdown of the two functions:

1. $f(x) = |\sin(x)|$:

   • This graph will have the same shape as the sine function, but all the negative parts of the sine wave will be reflected above the x-axis, making all values non-negative.

2. $f(x) = \sin(|x|)$:

   • This function is the sine of the absolute value of $x$, meaning it mirrors the sine wave around the y-axis. For negative values of $x$, it will behave as if $x$ were positive.

Now let's compare the graphs:

• Left graph: The sine wave looks mirrored around the x-axis, with all negative values made positive. This matches the behavior of $f(x) = |\sin(x)|$.

• Right graph: The graph is symmetric around the y-axis, which is the characteristic of $f(x) = \sin(|x|)$.

Conclusion:

• Left graph corresponds to $f(x) = |\sin(x)|$.
• Right graph corresponds to $f(x) = \sin(|x|)$.

Would you like further details on these functions or any follow-up questions?

Here are 5 related questions to expand your understanding:

1. How does the graph of $f(x) = \sin(x)$ differ from $f(x) = |\sin(x)|$?
2. What is the period of $f(x) = \sin(|x|)$?
3. How would you graph $f(x) = \cos(|x|)$?
4. What happens to $f(x) = \sin(x)$ when the absolute value is applied to both $x$ and $\sin(x)$?
5. Can you explain how symmetry around the x-axis and y-axis affects the graph?

Tip: When analyzing absolute value functions, observe how they alter both the negative and positive portions of a graph, reflecting or mirroring different parts.