To find the inverse of the function [tex]\( y = x^2 + 4x + 4 \)[/tex], we follow these steps:
1. Restate the function:
[tex]\[ y = x^2 + 4x + 4 \][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
Start by setting [tex]\( y \)[/tex] equal to the function:
[tex]\[ y = x^2 + 4x + 4 \][/tex]
Rewrite it as:
[tex]\[ y = (x+2)^2 \][/tex]
3. Take the square root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ \sqrt{y} = x + 2 \quad \text{or} \quad -\sqrt{y} = x + 2 \][/tex]
4. Isolate [tex]\( x \)[/tex]:
From [tex]\( \sqrt{y} = x + 2 \)[/tex]:
[tex]\[ x = \sqrt{y} - 2 \][/tex]
From [tex]\( -\sqrt{y} = x + 2 \)[/tex]:
[tex]\[ x = -\sqrt{y} - 2 \][/tex]
5. List the inverse functions:
The inverse functions are:
[tex]\[ x = \sqrt{y} - 2 \][/tex]
[tex]\[ x = -\sqrt{y} - 2 \][/tex]
Therefore, the inverses of the function [tex]\( y = x^2 + 4x + 4 \)[/tex] are:
[tex]\[ \boxed{-\sqrt{y} - 2 \text{ and } \sqrt{y} - 2} \][/tex]
