Question

what equation represents the inverse of $y = (x + 3)^2 + 4$?
a. $y = \pm \sqrt{x - 3} - 4$
b. $y = \pm \sqrt{x - 4} - 3$
c. $y = (x - 3)^2 - 4$
d. $y = (x - 4)^2 - 3

Explanation:

Step1: Swap x and y

To find the inverse of a function, we first swap the roles of \( x \) and \( y \) in the original equation. So starting with \( y=(x + 3)^{2}+4 \), we get \( x=(y + 3)^{2}+4 \).

Step2: Solve for y

First, isolate the squared term. Subtract 4 from both sides:
\( x-4=(y + 3)^{2} \)
Then, take the square root of both sides. Remember that when we take the square root of a number, we have to consider both the positive and negative roots:
\( \pm\sqrt{x - 4}=y + 3 \)
Finally, subtract 3 from both sides to solve for \( y \):
\( y=\pm\sqrt{x - 4}-3 \)

Answer:

B. \( y=\pm\sqrt{x - 4}-3 \)