To address this question, let’s analyze the function requirements step by step:

1. Domain

The function's domain is:
$[-5, 0) \cup (0, \infty)$
This means:

• The function is defined on two disjoint intervals: $[-5, 0)$ (from $-5$ to $0$, not including $0$) and $(0, \infty)$ (from $0$ onward, not including $0$).
• $x = 0$ is not in the domain.

2. Continuity

The function is continuous on its domain, so:

• No jumps, breaks, or holes within $[-5, 0)$ and $(0, \infty)$.
• However, the function is not defined at $x = 0$, so no continuity there.

3. Symmetry Condition

$f(-5) = f(5)$:

• This implies that at $x = -5$ and $x = 5$, the function values are equal.

4. Limit Conditions

• At $x \to 5$:
  $\lim_{x \to 5} f(x) = 6$
  As $x$ approaches $5$, the function value approaches $6$.

• At $x \to 0^-$ (from the left of $0$):
  $\lim_{x \to 0^-} f(x) = \infty$
  This indicates a vertical asymptote at $x = 0$ from the left side.

• At $x \to 0^+$ (from the right of $0$):
  $\lim_{x \to 0^+} f(x) = 0$
  The function value approaches $0$ as $x$ nears $0$ from the right side.

• As $x \to \infty$:
  $\lim_{x \to \infty} f(x) = -\infty$
  The function decreases unboundedly as $x$ increases without limit.

5. Graph Intercepts the x-Axis

The final question asks why the graph must intersect the $x$-axis, meaning there exists a point $c$ such that $f(c) = 0$.

• As $x \to 0^+$, the function approaches $0$, so it gets arbitrarily close to the x-axis.
• Since the function is continuous on $(0, \infty)$ and decreases to $-\infty$ as $x \to \infty$, by the Intermediate Value Theorem:
  • On some interval $(a, \infty)$, where $a$ is close to $0$, the function must cross the $x$-axis at least once.
  • This guarantees a $c > 0$ where $f(c) = 0$.

6. Summary of Key Points

The function’s graph:

• Has a vertical asymptote at $x = 0$.
• Approaches $6$ at $x = 5$.
• Decreases to $-\infty$ as $x \to \infty$.
• Must cross the $x$-axis due to the behavior described by the Intermediate Value Theorem.

Would you like me to sketch the graph or clarify any step further?

5 Related Questions:

1. What is the Intermediate Value Theorem, and how does it ensure $f(c) = 0$?
2. How does a vertical asymptote affect the continuity of a function at $x = 0$?
3. Can there be more than one $x$-intercept for the function described above?
4. How does the limit $\lim_{x \to \infty} f(x) = -\infty$ affect the graph?
5. Why does the condition $f(-5) = f(5)$ suggest symmetry at those points?

Tip:

When analyzing limit behavior at critical points (like $0$ here), pay close attention to the direction of approach (left-hand and right-hand limits).