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.

Explore how to determine the domain of the function g(t) = (t^2 - 1) / (t + 1) with step-by-step analysis. Learn about rational functions and the quotient rule in this detailed mathematical explanation. Suitable for Grades 11-12.

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