To determine which equation is the inverse of [tex]\( 5y + 4 = (x + 3)^2 + \frac{1}{2} \)[/tex], we need to manipulate the original equation and compare it to the given options.
First, let's solve the given equation for [tex]\( y \)[/tex]:
[tex]\[ 5y + 4 = (x + 3)^2 + \frac{1}{2} \][/tex]
Subtract 4 from both sides:
[tex]\[ 5y = (x + 3)^2 + \frac{1}{2} - 4 \][/tex]
Simplify the constant term on the right side:
[tex]\[ \frac{1}{2} - 4 = \frac{1}{2} - \frac{8}{2} = -\frac{7}{2} \][/tex]
Thus, the equation becomes:
[tex]\[ 5y = (x + 3)^2 - \frac{7}{2} \][/tex]
Divide both sides by 5 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{(x + 3)^2 - \frac{7}{2}}{5} \][/tex]
Now let's consider the given options and see which one can be manipulated to match the above form.
1. [tex]\( y = \frac{1}{5}x^2 + \frac{6}{5}x + \frac{11}{10} \)[/tex]
2. [tex]\( y = 3 \pm \sqrt{5x + \frac{7}{2}} \)[/tex]
3. [tex]\( -5y - 4 = -(x + 3)^2 - \frac{1}{2} \)[/tex]
4. [tex]\( y = -3 \pm \sqrt{5x + \frac{7}{2}} \)[/tex]
Examining Option 3, we have:
[tex]\[ -5y - 4 = -(x + 3)^2 - \frac{1}{2} \][/tex]
If we multiply both sides by -1 to simplify, we get:
[tex]\[ 5y + 4 = (x + 3)^2 + \frac{1}{2} \][/tex]
This matches exactly the given equation. Therefore, the correct option is Option 3:
[tex]\[ -5 y - 4 = -(x + 3)^2-\frac{1}{2} \][/tex]
Hence, the inverse equation in the list is:
[tex]\[ -5y - 4 = -(x + 3)^2 - \frac{1}{2} \][/tex]
So the answer is:
[tex]\[ 3 \][/tex]
