Find the inverse of the function.

[tex]y = x^2 + 4x + 4[/tex]

Possible answers:

A. [tex]y = \sqrt{\frac{1}{2}x + 2}[/tex]
B. [tex]y = \pm \sqrt{\frac{1}{2}x + 2}[/tex]
C. [tex]y = \pm \sqrt{x} - 2[/tex]
D. [tex]y = \sqrt{x} - 2[/tex]



Answer :

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]