Enumerate:  1. f(x) = x² + 5x + 6  where x = 1/(x-1)  2. f(x) square root of x² + 9x + 9  where x = 5x+1  3. f(x) = 1/(x² + 2x - 35) where x = x - 1

Let's evaluate each function with the given expressions for \( x \).

### 1. \( f(x) = x^2 + 5x + 6 \) where \( x = \frac{1}{x - 1} \)

Substitute \( x = \frac{1}{x - 1} \) into the function:

\[
f\left(\frac{1}{x - 1}\right) = \left(\frac{1}{x - 1}\right)^2 + 5\left(\frac{1}{x - 1}\right) + 6
\]

Simplify each term:
\[
\left(\frac{1}{x - 1}\right)^2 = \frac{1}{(x - 1)^2}
\]
\[
5\left(\frac{1}{x - 1}\right) = \frac{5}{x - 1}
\]

So,
\[
f\left(\frac{1}{x - 1}\right) = \frac{1}{(x - 1)^2} + \frac{5}{x - 1} + 6
\]

### 2. \( f(x) = \sqrt{x^2 + 9x + 9} \) where \( x = 5x + 1 \)

Substitute \( x = 5x + 1 \) into the function:

First, solve for \( x \):
\[
x = 5x + 1
\]
\[
x - 5x = 1
\]
\[
-4x = 1
\]
\[
x = -\frac{1}{4}
\]

Now, substitute \( x = -\frac{1}{4} \) into \( f(x) \):
\[
f\left(-\frac{1}{4}\right) = \sqrt{\left(-\frac{1}{4}\right)^2 + 9\left(-\frac{1}{4}\right) + 9}
\]
\[
= \sqrt{\frac{1}{16} - \frac{9}{4} + 9}
\]

To simplify,
\[
f\left(-\frac{1}{4}\right) = \sqrt{\frac{1}{16} - \frac{36}{16} + \frac{144}{16}} = \sqrt{\frac{109}{16}} = \frac{\sqrt{109}}{4}
\]

### 3. \( f(x) = \frac{1}{x^2 + 2x - 35} \) where \( x = x - 1 \)

This substitution simplifies as follows:

Since \( x = x - 1 \) is a contradiction, the value of \( x \) remains undefined, so the function does not change and remains \( f(x) = \frac{1}{x^2 + 2x - 35} \).

Let me know if you want more details or have any questions.

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**Related questions:**
1. How do you simplify a quadratic expression?
2. What are the steps to solve a square root function?
3. Can you explain the process of substituting values into a function?
4. What does it mean for a function to be undefined?
5. How do you solve a contradiction in algebra?
6. Can you derive the roots of a quadratic equation?
7. What is the significance of the discriminant in quadratic equations?
8. How do you handle complex numbers in square roots?

**Tip:** When dealing with contradictions, it's essential to check the original problem for any restrictions or possible errors in the assumptions made.

Learn how to evaluate functions with given expressions including quadratic equations and square root functions. This problem involves substituting values into functions and understanding domain restrictions. Suitable for advanced high school students.

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