To find the domain of the function

$g(t) = \frac{t^2 - 1}{t + 1},$

we need to identify the values of $t$ for which the function is defined. This function is a rational function, which means it is a fraction with a polynomial in the numerator and another in the denominator.

Step-by-Step Analysis

1. Identify where the function is undefined:

   A rational function is undefined wherever its denominator is equal to zero. Therefore, we need to find the value(s) of $t$ that make the denominator zero.

   $t + 1 = 0.$

   Solving for $t$:

   $t = -1.$

   Thus, the function is undefined at $t = -1$.

2. Consider the numerator:

   The numerator is $t^2 - 1$. We don't need to worry about the numerator being zero because it does not make the function undefined; it only makes the value of the function zero at those points.

   However, notice that:

   $t^2 - 1 = (t - 1)(t + 1).$

   The factor $t + 1$ appears in both the numerator and the denominator. Thus, at $t = -1$, the expression $\frac{t^2 - 1}{t + 1}$ simplifies to:

   $g(t) = \frac{(t - 1)(t + 1)}{t + 1}.$

   For $t \neq -1$, we can cancel out the $t + 1$ terms:

   $g(t) = t - 1, \quad \text{for } t \neq -1.$

3. Determine the domain:

   The only value excluded from the domain is $t = -1$, where the function is undefined.

Conclusion

The domain of the function $g(t) = \frac{t^2 - 1}{t + 1}$ is all real numbers except $t = -1$. In interval notation, the domain is:

$\boxed{(-\infty, -1) \cup (-1, \infty)}.$

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

Here are 5 related questions to expand your understanding:

1. What is the range of the function $g(t) = \frac{t^2 - 1}{t + 1}$?
2. How would you graph the function $g(t) = \frac{t^2 - 1}{t + 1}$?
3. What are the horizontal and vertical asymptotes of the function $g(t)$?
4. How would you find the derivative of $g(t) = \frac{t^2 - 1}{t + 1}$ using the quotient rule?
5. What happens to $g(t)$ as $t$ approaches $-1$ from the left and the right?

Tip: When working with rational functions, always check for points where the denominator is zero to determine the domain.