https://file.aiquickdraw.com/mathbot/file/17273668257026vdk9v13.jpg

The problem asks to determine the domain of the rational function \( h(x) = \frac{7x^3 - 5x - 49}{5x^2 - 19} \), and we need to round the results to the thousandths place.

### Steps to find the domain:
1. The domain of a rational function is all real numbers except where the denominator is zero, since division by zero is undefined.

2. Set the denominator equal to zero and solve:
   \[
   5x^2 - 19 = 0
   \]
   \[
   5x^2 = 19
   \]
   \[
   x^2 = \frac{19}{5}
   \]
   \[
   x = \pm \sqrt{\frac{19}{5}}
   \]

3. Calculate the roots:
   \[
   x = \pm \sqrt{3.8}
   \]
   Using a calculator, \( \sqrt{3.8} \approx 1.949 \).

   So, \( x = \pm 1.949 \).

### Domain:
The domain of the function excludes these values \( x = -1.949 \) and \( x = 1.949 \). Therefore, the domain is:
\[
D = (-\infty, -1.949) \cup (1.949, \infty)
\]

From the options provided in the image, the correct answer should be the third one: 
\[
D = (-\infty, -1.949) \cup (1.949, \infty).
\]

Would you like further explanation on any step, or do you have any questions?

Here are 5 related questions you could explore:
1. What happens to the graph of the function near \( x = -1.949 \) and \( x = 1.949 \)?
2. How does the behavior of a rational function change when the degree of the numerator and denominator are different?
3. How would you find vertical and horizontal asymptotes for this function?
4. What is the impact of removing the discontinuities from the function on its graph?
5. Could a function like this have any other types of singularities, such as holes in its graph?

**Tip:** Always check for both vertical asymptotes and holes in rational functions by simplifying the expression and analyzing the denominator carefully.

Determine the domain of the rational function h(x) = (7x^3 - 5x - 49) / (5x^2 - 19), rounded to the thousandths place.

Learn how to find the domain of the rational function h(x) = (7x^3 - 5x - 49) / (5x^2 - 19) by solving for when the denominator equals zero and rounding results to the thousandths place. This problem involves rational functions and square root calculations. Suitable 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