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.