To find the inverse of the function
[tex]\[ y = x^2 + 4x + 4 \][/tex]
we need to express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex].
### Step-by-Step Solution
1. Rewrite the original function:
The given function is
[tex]\[ y = x^2 + 4x + 4 \][/tex]
2. Express the quadratic function in standard form:
Notice that the quadratic function [tex]\( x^2 + 4x + 4 \)[/tex] can be written as a perfect square:
[tex]\[ x^2 + 4x + 4 = (x + 2)^2 \][/tex]
So, the function becomes:
[tex]\[ y = (x + 2)^2 \][/tex]
3. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
To find the inverse, we need to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ y = (x + 2)^2 \][/tex]
Take the square root of both sides to solve for [tex]\( x + 2 \)[/tex]:
[tex]\[ \sqrt{y} = x + 2 \][/tex]
or
[tex]\[ -\sqrt{y} = x + 2 \][/tex]
4. Isolate [tex]\( x \)[/tex]:
Solving for [tex]\( x \)[/tex], we get two solutions:
[tex]\[ x = \sqrt{y} - 2 \][/tex]
and
[tex]\[ x = -\sqrt{y} - 2 \][/tex]
Thus, the function [tex]\( y = x^2 + 4x + 4 \)[/tex] has two inverse functions:
[tex]\[ x = \sqrt{y} - 2 \][/tex]
and
[tex]\[ x = -\sqrt{y} - 2 \][/tex]
So the inverse functions are:
[tex]\[ f^{-1}(y) = \sqrt{y} - 2 \][/tex]
and
[tex]\[ f^{-1}(y) = -\sqrt{y} - 2 \][/tex]
Finally, these can also be rewritten as:
[tex]\[ x = \sqrt{y - 2} - 2 \][/tex]
and
[tex]\[ x = -\sqrt{y - 2} - 2 \][/tex]
So, the inverse functions are:
[tex]\[ f^{-1}(y) = \sqrt{y - 2} - 2 \][/tex]
and
[tex]\[ f^{-1}(y) = -\sqrt{y - 2} - 2 \][/tex]
These results indicate two possible inverse functions, capturing the symmetrical nature of quadratic equations.
