Answer:
B
Step-by-step explanation:
We are given the function:
[tex]\displaystyle f(x) = x^2 + 12x + 36[/tex]
And we want to determine the value of:
[tex]\displaystyle f^{-1}(225)[/tex]
Let this value equal a. In other words:
[tex]\displaystyle f^{-1}(225) = a[/tex]
Then by the definition of inverse functions:
[tex]\displaystyle \text{If } f^{-1}(225) = a\text{, then } f(a) = 225[/tex]
Hence:
[tex]\displaystyle f(a) =225 = (a)^2 + 12(a) + 36[/tex]
Solve for a:
[tex]\displaystyle \begin{aligned} 225 &= a^2 + 12a + 36 \\ \\ a^2 + 12a -189 &= 0 \\ \\ (a + 21)(a-9) &= 0\end{aligned}[/tex]
By the Zero Product Property:
[tex]\displaystyle a + 21 = 0 \text{ or } a - 9 = 0[/tex]
Hence:
[tex]\displaystyle a = -21 \text{ or } a = 9[/tex]
Thus, f(9) = 225. Consequently, f⁻¹(225) = 9.
In conclusion, our answer is B.
