Question 29 (1 point) Which of the following functions is the correct inverse for the function $$ f(x)=x^{2}+2 x+4 $$ ? A) $$ f^{-1}(x)=-4 \pm \sqrt{x} $$ B) $$ f^{-1}(x)=-2 \pm \sqrt{x} $$ C) $$ f^{-1}(x)=4 \pm \sqrt{x} $$ D) $$ f^{-1}(x)=2 \pm \sqrt{x} $$

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To find the inverse of a function, we need to switch the roles of $$ x $$ and $$ y $$ and then solve for the new $$ y $$.

Given function: $$ f(x) = x^{2} + 2x + 4 $$

Let's denote the inverse function as $$ f^{-1}(x) $$.

1. Replace $$ f(x) $$ with $$ y $$:
   $$ y = x^{2} + 2x + 4 $$

2. Swap $$ x $$ and $$ y $$:
   $$ x = y^{2} + 2y + 4 $$

3. Solve for $$ y $$:
   $$ x = y^{2} + 2y + 4 $$
   $$ y^{2} + 2y + 4 - x = 0 $$

Now, we need to solve this quadratic equation to find the inverse function. Let's solve it using the quadratic formula.
Solve the equation by following steps:
- step0: Solve for $$y$$:
  $$y^{2}+2y+4-x=0$$
- step1: Solve using the quadratic formula:
  $$y=\frac{-2\pm \sqrt{2^{2}-4\left(4-x\right)}}{2}$$
- step2: Simplify the expression:
  $$y=\frac{-2\pm \sqrt{-12+4x}}{2}$$
- step3: Simplify the expression:
  $$y=\frac{-2\pm 2\sqrt{-3+x}}{2}$$
- step4: Separate into possible cases:
  $$\begin{align}&y=\frac{-2+2\sqrt{-3+x}}{2}\&y=\frac{-2-2\sqrt{-3+x}}{2}\end{align}$$
- step5: Simplify the expression:
  $$\begin{align}&y=-1+\sqrt{-3+x}\&y=\frac{-2-2\sqrt{-3+x}}{2}\end{align}$$
- step6: Simplify the expression:
  $$\begin{align}&y=-1+\sqrt{-3+x}\&y=-1-\sqrt{-3+x}\end{align}$$
The solutions to the quadratic equation are:
$$ y = -1 + \sqrt{-3+x} $$
$$ y = -1 - \sqrt{-3+x} $$

Therefore, the correct inverse function for $$ f(x) = x^{2} + 2x + 4 $$ is:
$$ f^{-1}(x) = -1 + \sqrt{-3+x} $$
$$ f^{-1}(x) = -1 - \sqrt{-3+x} $$

Comparing these results with the given options:
A) $$ f^{-1}(x) = -4 \pm \sqrt{x} $$
B) $$ f^{-1}(x) = -2 \pm \sqrt{x} $$
C) $$ f^{-1}(x) = 4 \pm \sqrt{x} $$
D) $$ f^{-1}(x) = 2 \pm \sqrt{x} $$

The correct inverse function is $$ f^{-1}(x) = -1 \pm \sqrt{-3+x} $$, which is not listed in the options provided. Therefore, none of the given options are correct.
None of the given options are correct.

Question 29 (1 point)
Which of the following functions is the correct inverse for the function ?
A)
B)
C)
D)

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Question 29 (1 point) Which of the following functions is the correct inverse for the function ? A) B) C) D)