Find the domain and range of the inverse of f(x) = x²-6x+5 with domain restriction (x ∈ R 0 < x <3}.

To find the domain and range of the inverse of the function \( f(x) = x^2 - 6x + 5 \) with the domain restriction \( 0 < x < 3 \), we can follow these steps:

### 1. **Determine the range of the given function \( f(x) \):**
   - The function is a quadratic equation of the form \( f(x) = ax^2 + bx + c \), where \( a = 1 \), \( b = -6 \), and \( c = 5 \).
   - First, complete the square to express the function in vertex form.

     \[
     f(x) = x^2 - 6x + 5
     \]

     To complete the square:

     \[
     f(x) = (x^2 - 6x + 9) - 9 + 5 = (x - 3)^2 - 4
     \]

     So, 

     \[
     f(x) = (x - 3)^2 - 4
     \]

   - The vertex of the parabola is at \( x = 3 \), \( y = -4 \). Since the parabola opens upwards and the domain restriction is \( 0 < x < 3 \), the range of \( f(x) \) when \( 0 < x < 3 \) corresponds to \( -4 < f(x) < 5 \).

### 2. **Find the inverse function \( f^{-1}(x) \):**
   - To find the inverse, first, solve for \( x \) in terms of \( y \):

     \[
     y = (x - 3)^2 - 4
     \]

     Add 4 to both sides:

     \[
     y + 4 = (x - 3)^2
     \]

     Take the square root of both sides:

     \[
     \sqrt{y + 4} = x - 3
     \]

     Therefore,

     \[
     x = \sqrt{y + 4} + 3
     \]

     The inverse function is:

     \[
     f^{-1}(x) = \sqrt{x + 4} + 3
     \]

### 3. **Determine the domain and range of the inverse function \( f^{-1}(x) \):**
   - **Domain of \( f^{-1}(x) \):** The domain of the inverse function is the range of the original function. Since the range of \( f(x) \) was \( -4 < f(x) < 5 \), the domain of \( f^{-1}(x) \) is \( -4 < x < 5 \).
   - **Range of \( f^{-1}(x) \):** The range of the inverse function corresponds to the original domain \( 0 < x < 3 \). So, the range of \( f^{-1}(x) \) is \( 3 < f^{-1}(x) < 5 \).

### **Final Answer:**
- **Domain of \( f^{-1}(x) \):** \( -4 < x < 5 \)
- **Range of \( f^{-1}(x) \):** \( 3 < y < 5 \)

Would you like more details on any part of this solution?

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**Related Questions:**
1. How would the inverse function change if the domain restriction was \( x \geq 3 \)?
2. Can you explain how to graph the function \( f(x) = x^2 - 6x + 5 \) and its inverse?
3. What would be the domain and range of \( f^{-1}(x) \) if the original function had no domain restriction?
4. How do you find the inverse of a function that isn't one-to-one?
5. What is the significance of completing the square in finding the inverse of a quadratic function?

**Tip:** When dealing with quadratic functions, completing the square is a powerful technique for identifying key features like the vertex, which aids in finding the inverse.

Discover how to determine the domain and range of the inverse function for f(x) = x²-6x+5 with a domain restriction of 0 < x < 3. This problem involves quadratic equations and inverse function concepts, suitable for advanced high school students.

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