find the inverse of g(x) and provide both the domain and range of g^-1(x).  g(x) = -4 sqrroot of 5-2x then +1

To find the inverse of the function \( g(x) = -4\sqrt{5 - 2x} + 1 \), we need to follow these steps:

### Step 1: Replace \( g(x) \) with \( y \)
\[
y = -4\sqrt{5 - 2x} + 1
\]

### Step 2: Swap \( x \) and \( y \) to find the inverse
\[
x = -4\sqrt{5 - 2y} + 1
\]

### Step 3: Solve for \( y \) (the inverse function)

1. Subtract 1 from both sides:
   \[
   x - 1 = -4\sqrt{5 - 2y}
   \]

2. Divide by -4:
   \[
   \frac{x - 1}{-4} = \sqrt{5 - 2y}
   \]

3. Square both sides to eliminate the square root:
   \[
   \left(\frac{x - 1}{-4}\right)^2 = 5 - 2y
   \]

4. Simplify the left side:
   \[
   \frac{(x - 1)^2}{16} = 5 - 2y
   \]

5. Subtract 5 from both sides:
   \[
   \frac{(x - 1)^2}{16} - 5 = -2y
   \]

6. Multiply by -1:
   \[
   5 - \frac{(x - 1)^2}{16} = 2y
   \]

7. Divide by 2:
   \[
   y = \frac{5 - \frac{(x - 1)^2}{16}}{2}
   \]

Thus, the inverse function \( g^{-1}(x) \) is:
\[
g^{-1}(x) = \frac{5}{2} - \frac{(x - 1)^2}{32}
\]

### Step 4: Find the domain and range of \( g^{-1}(x) \)

1. **Domain of \( g^{-1}(x) \):**
   The domain of \( g^{-1}(x) \) is the range of the original function \( g(x) \). Since \( g(x) = -4\sqrt{5 - 2x} + 1 \), the square root must be non-negative, meaning \( 5 - 2x \geq 0 \), or \( x \leq \frac{5}{2} \). Therefore, the domain of \( g(x) \) is \( (-\infty, \frac{5}{2}] \). The range of \( g(x) \), and hence the domain of \( g^{-1}(x) \), is \( (-\infty, 1] \).

2. **Range of \( g^{-1}(x) \):**
   The range of \( g^{-1}(x) \) is the domain of \( g(x) \), which is \( (-\infty, \frac{5}{2}] \).

### Final Answer:
- The inverse function is \( g^{-1}(x) = \frac{5}{2} - \frac{(x - 1)^2}{32} \).
- Domain of \( g^{-1}(x) \): \( (-\infty, 1] \)
- Range of \( g^{-1}(x) \): \( (-\infty, \frac{5}{2}] \)

Would you like more details or have any questions? 

Here are five related questions:
1. How do you verify if a function and its inverse are correct?
2. Can the inverse function have a different range than the original?
3. What happens if the function includes complex numbers?
4. How do the transformations in \( g(x) \) affect its inverse?
5. How does the domain restriction influence the inverse function?

**Tip:** The domain of a function restricts the allowable values of \( x \), while the range reflects the possible outputs or values of \( y \). Always check both when finding inverses.

find the inverse of g(x) and provide both the domain and range of g^-1(x). g(x) = -4 sqrt(5-2x) + 1

Discover how to find the inverse of the function g(x) = -4√(5 - 2x) + 1. This detailed guide covers inverse function theory, solving radical equations, and determining the domain and range of g^-1(x). Suitable for students in Grades 10-12.

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