To find the inverse of the relation \( y = x^2 + 5 \), we need to solve for \( x \) in terms of \( y \).
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Start with the equation: \[ y = x^2 + 5 \]
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Subtract 5 from both sides: \[ y - 5 = x^2 \]
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Take the square root of both sides. Remember that we take both the positive and negative roots: \[ x = \pm \sqrt{y - 5} \]
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To express the inverse relation in terms of \( y \), we interchange \( x \) and \( y \): \[ y = \pm \sqrt{x - 5} \]
Thus, the inverse relation is: \[ y = \sqrt{x - 5} \quad \text{and} \quad y = -\sqrt{x - 5} \]
However, because \( y = x^2 + 5 \) is not a one-to-one function for all \( x \) (it fails the horizontal line test), we usually restrict the domain of \( x \) to make the function one-to-one.
If we restrict \( x \) to non-negative values \( x \geq 0 \), the inverse simplifies to: \[ y = \sqrt{x - 5} \]
So, the final inverse of the relation \( y = x^2 + 5 \) (with the restriction \( x \geq 0 \)) is: \[ y = \sqrt{x - 5} \quad \text{for } x \geq 5 \]