The mathematical expression provided consists of two functions $f(x)$ and $g(x)$, with the operation $\frac{f}{g}$. Here's the breakdown:

Given: $f(x) = x^2 - 1$ $g(x) = x + 1$

Task: Simplify $\frac{f(x)}{g(x)}$

Step 1: Substitute $f(x)$ and $g(x)$:

$\frac{f(x)}{g(x)} = \frac{x^2 - 1}{x + 1}$

Step 2: Factorize the numerator:

The numerator $x^2 - 1$ is a difference of squares: $x^2 - 1 = (x - 1)(x + 1)$

So the expression becomes: $\frac{f(x)}{g(x)} = \frac{(x - 1)(x + 1)}{x + 1}$

Step 3: Simplify:

Cancel out $x + 1$ (for $x \neq -1$): $\frac{f(x)}{g(x)} = x - 1, \quad \text{for } x \neq -1$

Final Answer:

$\frac{f(x)}{g(x)} = x - 1 \quad \text{(valid for } x \neq -1 \text{)}$

Would you like a more detailed explanation or additional examples? Let me know!

Here are 5 related questions to consider:

1. What happens to the function at $x = -1$ (undefined points)?
2. How do you graph the simplified function $\frac{f(x)}{g(x)}$?
3. What is the domain of $\frac{f(x)}{g(x)}$?
4. How would you handle similar rational expressions with higher degrees?
5. How do you check your work when simplifying rational expressions?

Tip: Always check the domain of a rational function before and after simplification, as certain values might make the denominator zero.