To determine if the two given functions \( y = x^2 + 5 \) and \( y_2 = x - 5 \) are inverses of each other, we need to check if the output of one function can be transformed back to the original input of the other function.
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Finding the inverse of \( y = x^2 + 5 \):
- Start with \( y = x^2 + 5 \)
- Solve for \( x \): \[ y - 5 = x^2 \ x = \sqrt{y - 5} \text{ or } x = -\sqrt{y - 5} \]
The inverse is not a single function; it's two parts due to the square root.
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Finding the output of the second function \( y_2 = x - 5 \): To check if they are true inverses of one another, we would plug \( y_2 \) into \( y \):
Substitute \( x = y_2 + 5 \) into \( y = x^2 + 5 \):
\[ y = (y_2 + 5)^2 + 5 \]
Expanding this yields: \[ y = (x - 5 + 5)^2 + 5 \ y = (x)^2 + 5 \ y = x^2 + 5 \]
This shows that \( y_2 = x - 5 \) is not the output or input necessary to return to \( x \).
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Therefore, checking if the ordered pairs are reflections across the line \( y = x \): To check if they are reflections of each other across the line \( y = x \), you should swap \( x \) and \( y \) in both functions.
The reflections do not yield the same outputs for the inputs, indicating that they do not serve as true inverses.
Final Conclusion:
The correct response is: No, they are not inverses because all corresponding ordered pairs are reflections of one another across the line \( y = x \).