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The image you've uploaded shows a problem where you need to determine the correct trigonometric function given certain inputs. 

The equation given is:

\[
f(x) = \sin^{-1}\left(\frac{1}{\sqrt{2}} \right)
\]

The options seem to ask you to match \( x = -\frac{\pi}{4} \), \( x = \frac{\pi}{4} \), or \( x = -\frac{\pi}{2} \) with \( f(x) \).

### To solve this:
1. The equation provided is \( f(x) = \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) \).
   
   We know that:
   
   \[
   \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4}
   \]
   because \( \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \).
   
2. Therefore, \( x = \frac{\pi}{4} \) is the correct answer.

Thus, the correct match is \( x = \frac{\pi}{4} \).

Would you like more details or have any questions about this? Here are some follow-up questions related to this topic:

1. How can you determine the values of inverse trigonometric functions for different inputs?
2. What is the domain and range of the \( \sin^{-1}(x) \) function?
3. How would you find the \( \sin^{-1}(x) \) for values not commonly found on the unit circle?
4. Can you describe the relationship between \( \sin(x) \) and \( \sin^{-1}(x) \)?
5. How would the problem change if the equation was \( f(x) = \cos^{-1}\left(\frac{1}{\sqrt{2}}\right) \)?

**Tip:** Remember that the inverse sine function, \( \sin^{-1}(x) \), returns an angle within the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \).

Given the trigonometric function f(x) = sin^(-1)(1/√2), determine which value of x matches the function.

Understand how to solve the trigonometric equation f(x) = sin^(-1)(1/√2) by matching it with the correct value of x. This problem covers key concepts in trigonometry, including the inverse sine function. Suitable for high school students in Grades 9-12.

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