To determine if the functions \( y = x^2 + 5 \) and \( y^2 = x - 5 \) are inverses of each other, we need to check if the ordered pairs (x, y) from the first equation correspond to (y, x) in the second equation, typically shown by the line \( y = x \).
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First function: \( y = x^2 + 5 \)
- If you have an ordered pair \( (a, b) \), then plugging in \( a \) into this function gives \( b = a^2 + 5 \).
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Second function: \( y^2 = x - 5 \)
- Rearranging gives \( y = \sqrt{x - 5} \) (only considering non-negative outputs for simplicity here).
- For ordered pair \( (b, a) \), plugging in \( b \) produces \( a^2 = b - 5 \).
To check if these functions are inverses, we need to see if swapping x and y gives us consistent results.
- If we substitute \( y = a^2 + 5 \) into \( y^2 = x - 5 \):
- Substitute \( b = a^2 + 5 \) to find \( a^2 \):
- \( (a^2 + 5)^2 = a - 5 \) is generally not true for all x and y coordinates.
- Substitute \( b = a^2 + 5 \) to find \( a^2 \):
To state whether they are inverses:
- They are not inverses because the equation \( y = x^2 + 5 \) does not yield an output for every possible x value that can be reflected back to the inputs of \( y^2 = x - 5 \). The graphical representation would reveal that they are not symmetric about the line \( y = x \).
So the correct response to the question would be:
No, they are not inverses because all corresponding ordered pairs are reflections of one another across the line \( y = -x \).