find the inverse function for g(x)= -2 sqrt1/3(x+1)  +4

To find the inverse of the function \( g(x) = -2 \sqrt{\frac{1}{3}(x + 1)} + 4 \), follow these steps:

### Step 1: Replace \( g(x) \) with \( y \)
This step is to help with the algebra.
\[
y = -2 \sqrt{\frac{1}{3}(x + 1)} + 4
\]

### Step 2: Swap \( x \) and \( y \)
To find the inverse function, interchange \( x \) and \( y \).
\[
x = -2 \sqrt{\frac{1}{3}(y + 1)} + 4
\]

### Step 3: Isolate the square root term
Subtract 4 from both sides:
\[
x - 4 = -2 \sqrt{\frac{1}{3}(y + 1)}
\]

Divide both sides by -2:
\[
\frac{x - 4}{-2} = \sqrt{\frac{1}{3}(y + 1)}
\]

Simplify the left side:
\[
\frac{4 - x}{2} = \sqrt{\frac{1}{3}(y + 1)}
\]

### Step 4: Eliminate the square root
Square both sides to remove the square root:
\[
\left(\frac{4 - x}{2}\right)^2 = \frac{1}{3}(y + 1)
\]

Expand the left side:
\[
\frac{(4 - x)^2}{4} = \frac{1}{3}(y + 1)
\]

### Step 5: Solve for \( y \)
Multiply both sides by 3 to eliminate the fraction on the right:
\[
\frac{3(4 - x)^2}{4} = y + 1
\]

Subtract 1 from both sides:
\[
y = \frac{3(4 - x)^2}{4} - 1
\]

### Step 6: Write the inverse function
The inverse function is:
\[
g^{-1}(x) = \frac{3(4 - x)^2}{4} - 1
\]

This is the inverse of the given function.

Do you want details on any of these steps or have any questions?

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Here are 5 related questions to explore:

1. How do you check if two functions are inverses of each other?
2. What are the domain and range of the inverse function \( g^{-1}(x) \)?
3. Can you find the derivative of the inverse function \( g^{-1}(x) \)?
4. How do transformations (shifts, stretches) affect the inverse of a function?
5. What is the significance of the inverse function in real-world applications?

**Tip:** Always verify if the inverse function works by substituting it back into the original equation.

find the inverse function for g(x)= -2 sqrt1/3(x+1) +4

Discover how to find the inverse of g(x) = -2√(1/3)(x+1) + 4 through a detailed step-by-step process. This solution involves algebraic manipulation, square roots, and inverse function concepts, making it ideal for high school students in Grades 10-12.

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