Question

Write the inverse of the given relation.
y=x^2+5

Answer 1

To find the inverse of the relation \( y = x^2 + 5 \), we need to solve for \( x \) in terms of \( y \).

1. Start with the equation:
\[
y = x^2 + 5
\]

2. Subtract 5 from both sides:
\[
y - 5 = x^2
\]

3. Take the square root of both sides. Remember that we take both the positive and negative roots:
\[
x = \pm \sqrt{y - 5}
\]

4. 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:
\[
y = \sqrt{x - 5} \quad \text{for } x \geq 5
\]