$$ f\left(x\right)=\sqrt{\left(x-2\right)^2,\text{ }g\left(x\right)=x^2-4,\left(\right.}g $$/$$ f)\left(x\right)= $$F(x)/g(x)

It looks like you're working with functions $ f(x) $ and $ g(x) $, and you want to find the expression for $ \frac{f(x)}{g(x)} $.

However, the function $ f(x) $ appears to have a formatting issue. Let's clarify it:

1. **Assuming $ f(x) = \sqrt{(x-2)^2} $**:
   - $ f(x) $ simplifies to $ |x-2| $ because $ \sqrt{(x-2)^2} $ is the absolute value of $ x-2 $.

2. **Given $ g(x) = x^2 - 4 $**:
   - This can be factored as $ g(x) = (x-2)(x+2) $.

Now, the expression for $ \frac{f(x)}{g(x)} $ becomes:

\[
\frac{f(x)}{g(x)} = \frac{|x-2|}{(x-2)(x+2)}
\]

This expression simplifies depending on the value of $ x $:

- **If $ x > 2 $**, $ |x-2| = x-2 $, so:
  \[
  \frac{f(x)}{g(x)} = \frac{x-2}{(x-2)(x+2)} = \frac{1}{x+2}
  \]

- **If $ x < 2 $**, $ |x-2| = -(x-2) = 2-x $, so:
  \[
  \frac{f(x)}{g(x)} = \frac{2-x}{(x-2)(x+2)} = \frac{-(x-2)}{(x-2)(x+2)} = \frac{-1}{x+2}
  \]

- **If $ x = 2 $**, $ g(x) = 0 $ and $ f(x) = 0 $, making the expression undefined.

### Final Answer:
\[
\frac{f(x)}{g(x)} = \begin{cases} 
\frac{1}{x+2}, & \text{if } x > 2 \\
\frac{-1}{x+2}, & \text{if } x < 2 \\
\text{undefined}, & \text{if } x = 2
\end{cases}
\]

Would you like more details on any of the steps, or do you have any other questions?

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**Relative Questions:**

1. How does the absolute value function affect the behavior of $ f(x) $?
2. What are the implications of the function being undefined at $ x = 2 $?
3. How does the sign of $ x-2 $ affect the simplification of the fraction $ \frac{f(x)}{g(x)} $?
4. What is the domain of the function $ \frac{f(x)}{g(x)} $?
5. How can the behavior of $ \frac{f(x)}{g(x)} $ near $ x = 2 $ be analyzed?

**Tip:** Always check for points where the denominator could be zero when simplifying rational expressions!

Explore how to simplify rational expressions involving absolute values and rational functions. This detailed explanation covers scenarios where functions are undefined and the impact of absolute values on mathematical expressions. Suitable for Grades 10-12 students.

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