To determine the inverse of the equation [tex]\(5y + 4 = (x + 3)^2 + \frac{1}{2}\)[/tex], we need to isolate [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex].
### Step-by-Step Solution:
1. Start with the given equation:
[tex]\[
5y + 4 = (x + 3)^2 + \frac{1}{2}
\][/tex]
2. Isolate [tex]\(5y\)[/tex]:
[tex]\[
5y = (x + 3)^2 + \frac{1}{2} - 4
\][/tex]
3. Simplify the expression on the right-hand side:
[tex]\[
5y = (x + 3)^2 + \frac{1}{2} - 4
\][/tex]
[tex]\[
5y = (x + 3)^2 + \frac{1}{2} - \frac{8}{2}
\][/tex]
[tex]\[
5y = (x + 3)^2 - \frac{7}{2}
\][/tex]
4. Solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{1}{5} \left((x + 3)^2 - \frac{7}{2}\right)
\][/tex]
[tex]\[
y = \frac{1}{5} (x + 3)^2 - \frac{7}{10}
\][/tex]
5. Expand [tex]\(y\)[/tex] (if needed):
We can expand [tex]\((x + 3)^2 = x^2 + 6x + 9\)[/tex]:
[tex]\[
y = \frac{1}{5} (x^2 + 6x + 9) - \frac{7}{10}
\][/tex]
[tex]\[
y = \frac{1}{5} x^2 + \frac{6}{5} x + \frac{9}{5} - \frac{7}{10}
\][/tex]
[tex]\[
y = \frac{1}{5} x^2 + \frac{6}{5} x + \frac{18}{10} - \frac{7}{10}
\][/tex]
[tex]\[
y = \frac{1}{5} x^2 + \frac{6}{5} x + \frac{11}{10}
\][/tex]
Hence, the equation for the inverse function is:
[tex]\[
y = \frac{1}{5} x^2 + \frac{6}{5} x + \frac{11}{10}
\][/tex]
Thus, the correct answer from the given options is:
[tex]\[
y = \frac{1}{5} x^2 + \frac{6}{5} x + \frac{11}{10}
\][/tex]
