To find the inverse of the function [tex]\( y = x^2 - 12 \)[/tex], we need to follow these steps:
1. Start with the original function:
[tex]\[
y = x^2 - 12
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. This step is due to the definition of an inverse function, which exchanges the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = y^2 - 12
\][/tex]
3. Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]. First, isolate [tex]\( y^2 \)[/tex] by adding 12 to both sides:
[tex]\[
x + 12 = y^2
\][/tex]
4. Take the square root of both sides. Remember that taking the square root introduces both a positive and a negative solution:
[tex]\[
y = \pm \sqrt{x + 12}
\][/tex]
Thus, the inverse function is:
[tex]\[
y = \pm \sqrt{x + 12}
\][/tex]
Among the given options, the correct inverse function is:
[tex]\[ y = \pm \sqrt{x + 12} \][/tex]
