https://file.aiquickdraw.com/mathbot/file/17258681059800uffoh2t.jpg

Let's find the domain and range of the function \( G(t) = \frac{4}{t^2 - 1} \).

### Domain of \( G(t) \)

The domain of the function consists of all values of \( t \) for which the function is defined. The function \( G(t) \) is undefined when the denominator is zero, so we need to find the values of \( t \) that make the denominator zero:

\[
t^2 - 1 = 0
\]

Solving this equation:

\[
t^2 = 1 \implies t = \pm 1
\]

Therefore, the function is undefined at \( t = 1 \) and \( t = -1 \). So, the domain of \( G(t) \) in interval notation is:

\[
(-\infty, -1) \cup (-1, 1) \cup (1, \infty)
\]

### Range of \( G(t) \)

To find the range, let's consider the function behavior:

1. \( G(t) = \frac{4}{t^2 - 1} \), where \( t^2 - 1 \) can take any real value except zero.
2. As \( t^2 - 1 \) approaches 0 (near \( t = \pm 1 \)), \( G(t) \) approaches \( \pm \infty \).
3. As \( t^2 - 1 \) becomes very large (positively or negatively), \( G(t) \) approaches 0 but never actually reaches 0.

Thus, the function cannot output 0, and the range is all real numbers except 0:

\[
(-\infty, 0) \cup (0, \infty)
\]

Would you like a detailed explanation of any step, or do you have any other questions?

### Related Questions:
1. How would the domain change if the function was \( G(t) = \frac{4}{t^2 + 1} \)?
2. How do vertical asymptotes relate to the domain of a function?
3. How can we find the inverse of \( G(t) = \frac{4}{t^2 - 1} \), and what would be its domain?
4. What happens to the range if the function is \( G(t) = \frac{4}{t^2 - 2} \)?
5. How do you determine horizontal asymptotes of rational functions?

**Tip:** Always check for values that make the denominator zero when determining the domain of rational functions.

Find the domain and range of the function G(t) = 4 / (t^2 - 1).

Understand how to find the domain and range of the rational function G(t) = 4 / (t^2 - 1). This example explains the key concepts of rational functions, including handling undefined values at t = 1 and t = -1. Perfect for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation