The given function is:

\[
h(x) = \frac{7 + x^2}{x - 7}
\]

### Key Aspects to Analyze:

1. **Domain of the Function**:
   The function is undefined where the denominator equals zero. So, set \( x - 7 = 0 \):

   \[
   x = 7
   \]

   Therefore, the domain is \( x \in \mathbb{R} \) except \( x = 7 \).

2. **Vertical Asymptote**:
   Since the function is undefined at \( x = 7 \), there is a vertical asymptote at \( x = 7 \).

3. **Horizontal Asymptote**:
   To find the horizontal asymptote, analyze the behavior of the function as \( x \) approaches \( \infty \) or \( -\infty \). The highest degree term dominates, so:

   \[
   h(x) \approx \frac{x^2}{x} = x \text{ as } x \to \pm\infty.
   \]

   Therefore, there is **no** horizontal asymptote, but the function behaves linearly for large values of \( x \).

4. **Intercepts**:
   - **y-intercept**: Set \( x = 0 \):

     \[
     h(0) = \frac{7 + 0^2}{0 - 7} = \frac{7}{-7} = -1.
     \]

     So, the y-intercept is \( (0, -1) \).

   - **x-intercepts**: Set \( h(x) = 0 \):

     \[
     \frac{7 + x^2}{x - 7} = 0.
     \]

     The numerator must equal zero:

     \[
     7 + x^2 = 0 \implies x^2 = -7.
     \]

     Since \( x^2 = -7 \) has no real solutions, there are no x-intercepts.

### Do you want more details or have any questions?

---

Here are some related questions to deepen your understanding:

1. How does the graph of \( h(x) = \frac{7 + x^2}{x - 7} \) behave near the vertical asymptote at \( x = 7 \)?
2. What happens if we factor the function differently? Can it be simplified further?
3. What would change if the numerator had a linear term instead of \( x^2 \)?
4. How does the presence of \( x - 7 \) in the denominator impact the graph compared to a simpler rational function?
5. What are the potential applications of understanding such rational functions?

**Tip**: When analyzing rational functions, always check for asymptotes first—they provide a quick overview of the graph’s shape and behavior!

Explore the analysis of the rational function h(x) = (7 + x^2) / (x - 7), including its domain, vertical asymptote, lack of horizontal asymptote, and intercepts. Learn how to interpret the behavior of this function graphically and algebraically. Suitable for high school students studying advanced algebra and calculus concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation