To find the domain and range of the function 

\[
f(t) = \frac{8}{t^2 - 64},
\]

we will analyze it step by step.

### Domain:

The domain of a function is the set of all possible input values (in this case, values of \( t \)) that do not cause the function to be undefined.

1. The function \( f(t) \) will be undefined when the denominator is zero because division by zero is not allowed.
   
2. Set the denominator equal to zero to find these points:

   \[
   t^2 - 64 = 0.
   \]

3. Solve this equation:

   \[
   t^2 = 64 \implies t = \pm 8.
   \]

4. Therefore, the function is undefined at \( t = 8 \) and \( t = -8 \).

5. So, the domain of the function is all real numbers except \( t = 8 \) and \( t = -8 \).

   \[
   \text{Domain: } \{t \in \mathbb{R} \mid t \neq 8, t \neq -8\}.
   \]

### Range:

The range of a function is the set of all possible output values.

1. Consider the function form:

   \[
   f(t) = \frac{8}{t^2 - 64}.
   \]

2. The denominator can be rewritten as \( t^2 - 64 = (t - 8)(t + 8)

Explore how to determine the domain and range of the function 8/(t^2 - 64). This problem involves analyzing rational functions, understanding when the function is undefined, and identifying output ranges. Suitable for students in Grades 9-11.

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