Here's the step-by-step process to find the inverse of the given equation \(5y + 4 = (x + 3)^2 + \frac{1}{2}\):
1. Rewrite the equation by interchanging \(x\) and \(y\):
Original equation: \(5y + 4 = (x + 3)^2 + \frac{1}{2}\)
Swap \(x\) and \(y\):
[tex]\[5x + 4 = (y + 3)^2 + \frac{1}{2}\][/tex]
2. Solve for the new \(y\):
- Start by isolating \((y + 3)^2\) on one side of the equation:
[tex]\[5x + 4 - \frac{1}{2} = (y + 3)^2\][/tex]
Combine \(4\) and \(\frac{1}{2}\):
[tex]\[5x + 4 - \frac{1}{2} = (y + 3)^2\][/tex]
[tex]\[5x + \frac{8}{2} - \frac{1}{2} = (y + 3)^2\][/tex]
[tex]\[5x + \frac{7}{2} = (y + 3)^2\][/tex]
- Next, isolate \(y\) by taking the square root of both sides. Remember to include the \( \pm \) symbol as the square root can be both positive and negative:
[tex]\[y + 3 = \pm \sqrt{5x + \frac{7}{2}}\][/tex]
- Solve for \(y\):
[tex]\[y = -3 \pm \sqrt{5x + \frac{7}{2}}\][/tex]
The inverse equation is:
[tex]\[y = -3 \pm \sqrt{5x + \frac{7}{2}}\][/tex]
So, from the given options, the correct inverse equation is:
[tex]\[y = -3 \pm \sqrt{5x + \frac{7}{2}}\][/tex]
Therefore, the right answer is:
[tex]\(y = -3 \pm \sqrt{5x + \frac{7}{2}}\)[/tex]
