https://r2.erweima.ai/mathbot/image/17260821601241gcdzff4.jpg

The problem asks to determine the inverse function \( f^{-1} \) and its domain for the function \( f(x) = x^2 - 2x + 3 \), given that \( x \in (-3, 1) \).

### Step-by-Step Solution:

1. **Original Function:**
   \[
   f(x) = x^2 - 2x + 3
   \]
   We need to find the inverse \( f^{-1}(x) \). 

2. **Rewrite the function:**
   First, to simplify finding the inverse, we complete the square:
   \[
   f(x) = x^2 - 2x + 3
   \]
   To complete the square, take half of the coefficient of \( x \) (which is -2), square it, and add and subtract the result:
   \[
   f(x) = (x^2 - 2x + 1) + 2 = (x - 1)^2 + 2
   \]
   So, the function becomes:
   \[
   f(x) = (x - 1)^2 + 2
   \]

3. **Solve for \( x \):**
   To find the inverse, solve for \( x \) in terms of \( y \) (let \( f(x) = y \)):
   \[
   y = (x - 1)^2 + 2
   \]
   Subtract 2 from both sides:
   \[
   y - 2 = (x - 1)^2
   \]
   Take the square root of both sides:
   \[
   \sqrt{y - 2} = |x - 1|
   \]
   Solve for \( x \):
   \[
   x = 1 \pm \sqrt{y - 2}
   \]

4. **Determine the correct branch for the inverse:**
   Since \( x \in (-3, 1) \), we are restricted to the decreasing part of the parabola. This means we take the negative square root:
   \[
   x = 1 - \sqrt{y - 2}
   \]

5. **Final Inverse Function:**
   Thus, the inverse function is:
   \[
   f^{-1}(y) = 1 - \sqrt{y - 2}
   \]

6. **Domain of the Inverse Function:**
   The function \( f(x) = x^2 - 2x + 3 \) has a restricted domain \( x \in (-3, 1) \). For the inverse function, the domain of \( f^{-1}(y) \) corresponds to the range of \( f(x) \) for \( x \in (-3, 1) \).

   Evaluate the range by checking the values of \( f(x) \) at the endpoints of the interval:
   - \( f(-3) = (-3)^2 - 2(-3) + 3 = 9 + 6 + 3 = 18 \)
   - \( f(1) = (1)^2 - 2(1) + 3 = 1 - 2 + 3 = 2 \)

   Therefore, the range of \( f(x) \) is \( y \in [2, 18) \), which means the domain of \( f^{-1}(y) \) is:
   \[
   y \in [2, 18)
   \]

### Conclusion:
- The inverse function is \( f^{-1}(y) = 1 - \sqrt{y - 2} \).
- The domain of the inverse function is \( y \in [2, 18) \).

Let me know if you want further details or clarifications!

### Related Questions:
1. How do you derive the domain of a function from its inverse?
2. Why do we take the negative square root for the inverse function in this case?
3. How does completing the square help in finding the inverse of a quadratic function?
4. What would happen to the inverse function if the domain of \( f(x) \) were unrestricted?
5. How do we determine the range of a quadratic function given its domain?

### Tip:
Always verify if the function is one-to-one in the given domain before calculating its inverse.

Determine the inverse function f^{-1} and its domain for the function f(x) = x^2 - 2x + 3, given that x ∈ (-3, 1).

Learn how to find the inverse function of a quadratic function x^2 - 2x + 3 within the interval x ∈ (-3, 1). This detailed guide explains the process of completing the square and solving for the inverse, suitable for high school students learning advanced algebra topics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation