To determine whether the functions \( f(x) = x^2 \) and \( y = -f(x) = -x^2 \) are inverses of each other, we need to check if they reflect across the line \( y = x \).
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Check if they are inverses: For two functions to be inverses, the following must hold true:
- If \( (a, b) \) is on the graph of \( f \), then \( (b, a) \) must be on the graph of \( y = -f(x) \).
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Consider a pair: Take \( x = 1 \):
- \( f(1) = 1^2 = 1 \) → Corresponding ordered pair is \( (1, 1) \).
- For \( y = -f(x) \), calculate \( y = -(1^2) = -1 \) → Corresponding ordered pair is \( (1, -1) \).
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Check another pair: Take \( x = 2 \):
- \( f(2) = 2^2 = 4 \) → Corresponding ordered pair is \( (2, 4) \).
- For \( y = -f(x) \), calculate \( y = -(2^2) = -4 \) → Corresponding ordered pair is \( (4, -4) \).
From the examples above, you can see that \( f(1) = 1 \) gives a point \( (1, 1) \) and corresponds to the point \( (1, -1) \) on the graph of \( y = -f(x) \). Clearly, the resulting points do not match the reflection condition required for inverse functions across the line \( y = x \).
- Conclusion: Since there are instances where corresponding ordered pairs do not exhibit the required relationship to be inverses, we conclude that:
No, not all corresponding ordered pairs are reflections of one another across the line \( y = x \).