The function you provided is:

\[
F(x) = \frac{5x + 2}{x - 7}
\]

This is a rational function, where the numerator is \(5x + 2\) and the denominator is \(x - 7\).

### Key Properties:

1. **Domain**: The function is undefined when the denominator is zero. So, \(x - 7 = 0\) implies \(x = 7\). Therefore, the domain of \(F(x)\) is all real numbers except \(x = 7\).

   \[
   \text{Domain: } x \in \mathbb{R}, \, x \neq 7
   \]

2. **Vertical Asymptote**: The vertical asymptote occurs where the denominator is zero (which is the same as where the function is undefined). Therefore, there is a vertical asymptote at \(x = 7\).

3. **Horizontal Asymptote**: To find the horizontal asymptote, we compare the degrees of the polynomials in the numerator and the denominator.
   - The degree of the numerator \(5x + 2\) is 1.
   - The degree of the denominator \(x - 7\) is also 1.
   
   Since the degrees are equal, the horizontal asymptote is determined by the ratio of the leading coefficients:
   \[
   y = \frac{5}{1} = 5
   \]
   Therefore, the horizontal asymptote is \(y = 5\).

4. **Intercepts**:
   - **Y-intercept**: Set \(x = 0\) to find the y-intercept.
     \[
     F(0) = \frac{5(0) + 2}{0 - 7} = \frac{2}{-7} = -\frac{2}{7}
     \]
     So, the y-intercept is \(\left(0, -\frac{2}{7}\right)\).
   
   - **X-intercept**: Set the numerator equal to zero to find the x-intercept.
     \[
     5x + 2 = 0 \implies x = -\frac{2}{5}
     \]
     So, the x-intercept is \(\left(-\frac{2}{5}, 0\right)\).

Would you like a more detailed explanation of any part of this function?

Here are some related questions you might find helpful:
1. How do you determine slant asymptotes in rational functions?
2. What is the behavior of the function as \(x\) approaches the vertical asymptote?
3. How can you graph this rational function?
4. What are the steps to find the inverse of this function?
5. Can you explain how to factor the numerator and denominator in more complex rational functions?

**Tip:** Always check the domain of a function before performing operations like addition, subtraction, or composition with other functions.

Explore the rational function F(x) = (5x + 2) / (x - 7) with detailed explanations on its domain, asymptotes, and intercepts. Learn how to find the domain, vertical and horizontal asymptotes, and intercepts of this function. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation