Answer:
Option C: [tex]\displaystyle y = -3 \pm \sqrt{5 x + \frac{7}{2}[/tex].
Step-by-step explanation:
Typically, solving an equation means finding an expression of [tex]y[/tex] that is written in [tex]x[/tex]. However, this question is asking for the inverse of an equation in an x-y plane. Thus, start by solving for an expression of [tex]x[/tex] that is written in [tex]y[/tex]. After that, interchange [tex]x[/tex] and [tex]y[/tex] to obtain the equation of the inverse.
[tex]\displaystyle 5 y + 4 = (x + 3)^{2} + \frac{1}{2}[/tex].
[tex]\displaystyle (x + 3)^{2} + \frac{1}{2} = 5 y + 4[/tex].
Subtract 1/2 from both sides of this equation:
[tex]\displaystyle (x + 3)^{2} = 5 y + \frac{7}{2}[/tex].
Take the square root of both sides of this equation; note the plus-minus sign on the right-hand side of this equation:
[tex]\displaystyle (x + 3) = \pm(\5 y + \frac{7}{2})[/tex].
Subtract [tex]3[/tex] from both sides of this equation:
[tex]\displaystyle x = -3 \pm(\5 y + \frac{7}{2})[/tex].
Interchange [tex]x[/tex] and [tex]y[/tex] to obtain the equation for the inverse:
[tex]\displaystyle y = -3 \pm(\5 x + \frac{7}{2})[/tex].
