let g(x)=$$ \sqrt{x+2} $$ state the equation for g$$ ^{-1} $$(x) state the domain of g$$ ^{-1} $$(x) and graph g(x) and g$$ ^{-1} $$(x) and solve $$ {\displaylines\sqrt{x^2}-5=x+1\\ \placeholder{}} $$

Let's go through each part of your request:

### 1. Inverse of $ g(x) = \sqrt{x+2} $

To find the inverse, we swap $ x $ and $ y $ in the equation and solve for $ y $:
\[
g(x) = \sqrt{x + 2}
\]
Let $ y = \sqrt{x + 2} $. To find the inverse:
\[
x = \sqrt{y + 2}
\]
Square both sides:
\[
x^2 = y + 2
\]
Now solve for $ y $:
\[
y = x^2 - 2
\]
Thus, the inverse function is:
\[
g^{-1}(x) = x^2 - 2
\]

### 2. Domain of $ g^{-1}(x) $

The domain of $ g^{-1}(x) $ is the range of $ g(x) $. The function $ g(x) = \sqrt{x+2} $ is defined for $ x \geq -2 $, and since square roots are non-negative, $ g(x) \geq 0 $. Therefore, the range of $ g(x) $ is $ [0, \infty) $.

Hence, the domain of $ g^{-1}(x) = x^2 - 2 $ is $ [0, \infty) $.

### 3. Graph of $ g(x) $ and $ g^{-1}(x) $

We need to graph both $ g(x) = \sqrt{x+2} $ and $ g^{-1}(x) = x^2 - 2 $. The graph of an inverse function is a reflection of the original function over the line $ y = x $. 

### 4. Solve the equation

The equation to solve is:
\[
\sqrt{x^2} - 5 = x + 1
\]
First, simplify $ \sqrt{x^2} $. Since $ \sqrt{x^2} = |x| $, the equation becomes:
\[
|x| - 5 = x + 1
\]

Now, solve this equation for two cases:

#### Case 1: $ x \geq 0 $ 
In this case, $ |x| = x $, so the equation becomes:
\[
x - 5 = x + 1
\]
Subtract $ x $ from both sides:
\[
-5 = 1
\]
This is a contradiction, so there are no solutions in this case.

#### Case 2: $ x < 0 $
In this case, $ |x| = -x $, so the equation becomes:
\[
-x - 5 = x + 1
\]
Add $ x $ to both sides:
\[
-5 = 2x + 1
\]
Subtract 1 from both sides:
\[
-6 = 2x
\]
Now, divide by 2:
\[
x = -3
\]

Thus, the solution to the equation is $ x = -3 $.

Would you like me to plot the graphs or explain any part in more detail?

---

### Follow-up Questions
1. How do we verify the inverse function algebraically?
2. Why is $ g^{-1}(x) $ only defined for $ x \geq 0 $?
3. Can the solution $ x = -3 $ be verified by substitution?
4. What happens if you try to solve the equation $ |x| - 5 = x + 1 $ by squaring both sides directly?
5. How does the line $ y = x $ help in finding inverse function graphs?

### Tip:
When solving absolute value equations, always consider both the positive and negative cases separately.

Let g(x) = sqrt(x+2). State the equation for g^(-1)(x), state the domain of g^(-1)(x), and graph g(x) and g^(-1)(x). Solve sqrt(x^2) - 5 = x + 1.

Learn how to determine the inverse function of g(x) = sqrt(x+2) and its domain, graph both g(x) and its inverse, and solve the absolute value equation sqrt(x^2) - 5 = x + 1. Detailed explanations and step-by-step solutions are provided, suitable for students in Grades 10-12.

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