To find the inverse of the function [tex]\( y = x^2 + 4x + 4 \)[/tex], follow these steps:
1. Rewrite the function in terms of [tex]\( y \)[/tex]:
Given the function:
[tex]\[
y = x^2 + 4x + 4
\][/tex]
2. Express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
To find the inverse function, we need to solve the equation for [tex]\( x \)[/tex]. Start by setting [tex]\( y \)[/tex] equal to the right-hand side:
[tex]\[
y = x^2 + 4x + 4
\][/tex]
3. Convert the quadratic equation to standard form:
Rearrange the equation to isolate the quadratic term, making it:
[tex]\[
x^2 + 4x + (4 - y) = 0
\][/tex]
4. Solve the quadratic equation:
Use the quadratic formula, [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = 4 - y \)[/tex].
Substitute the values:
[tex]\[
x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot (4 - y)}}{2 \cdot 1}
\][/tex]
Simplify inside the square root:
[tex]\[
x = \frac{-4 \pm \sqrt{16 - 16 + 4y}}{2}
\][/tex]
[tex]\[
x = \frac{-4 \pm \sqrt{4y}}{2}
\][/tex]
5. Simplify the expression:
Simplify the square root and the fractions to get:
[tex]\[
x = \frac{-4 \pm 2\sqrt{y}}{2}
\][/tex]
[tex]\[
x = -2 \pm \sqrt{y}
\][/tex]
6. State the inverse functions:
The solutions indicate that there are two branches for the inverse function, giving us:
[tex]\[
x = -2 + \sqrt{y}
\][/tex]
and
[tex]\[
x = -2 - \sqrt{y}
\][/tex]
Hence, the inverse functions of [tex]\( y = x^2 + 4x + 4 \)[/tex] are:
[tex]\[
x = \sqrt{y} - 2 \quad \text{and} \quad x = -\sqrt{y} - 2
\][/tex]
