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

1. Rewrite the function:

First, let's rewrite $f(x)$ in a more convenient form by completing the square:

$f(x) = x^2 - 6x + 5$

Complete the square:

$f(x) = (x^2 - 6x + 9) - 9 + 5 = (x - 3)^2 - 4$

So the function becomes:

$f(x) = (x - 3)^2 - 4$

2. Determine the domain of the inverse function:

Since the original function $f(x)$ has the domain restriction $0 < x < 3$, this domain is the range of the inverse function.

3. Find the inverse function:

To find the inverse, we solve for $x$ in terms of $y$:

$y = (x - 3)^2 - 4$

$y + 4 = (x - 3)^2$

Take the square root of both sides:

$\sqrt{y + 4} = |x - 3|$

Since the domain of the original function is $0 < x < 3$, $x - 3$ is negative. Therefore:

$\sqrt{y + 4} = 3 - x$

$x = 3 - \sqrt{y + 4}$

Thus, the inverse function is:

$f^{-1}(y) = 3 - \sqrt{y + 4}$

4. Determine the range of the inverse function:

The range of the inverse function corresponds to the domain of the original function. Since the original function $f(x) = (x - 3)^2 - 4$ with $0 < x < 3$:

• At $x = 0$, $f(0) = (0 - 3)^2 - 4 = 9 - 4 = 5$.
• At $x = 3$, $f(3) = (3 - 3)^2 - 4 = 0 - 4 = -4$.

So, the range of the inverse function is $-4 < y < 5$.

5. Summary:

• Domain of the inverse function: $-4 < y < 5$
• Range of the inverse function: $0 < x < 3$

6. Graph the Inverse Function:

The graph of the inverse function $f^{-1}(y) = 3 - \sqrt{y + 4}$ can be plotted within the specified domain and range.

I can generate a graph of the inverse function if you would like. Would you like to see that?

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Questions:

1. How do you complete the square for a quadratic function?
2. What is the significance of the domain restriction when finding an inverse?
3. How do you determine the range of the inverse function?
4. How can you verify that a function is one-to-one before finding its inverse?
5. What happens to the graph of a function when you find its inverse?

Tip: Remember that when finding an inverse, it's crucial to ensure that the function is one-to-one on the specified domain to guarantee the existence of an inverse.