Rational Function X Intercepts 4C0651

1. **Problem Statement:** We are given a rational function graph with vertical asymptotes at $x = -2$ and $x = -1$, and a horizontal asymptote at $y = 0$. The graph has three branches: one in the top-right, one between $x = -2$ and $x = -1$ crossing downward through the center, and one in the bottom-left. We need to find the $x$-intercepts of this graph. 2. **Understanding the function:** A rational function with vertical asymptotes at $x = -2$ and $x = -1$ can be expressed as: $$f(x) = \frac{P(x)}{(x+2)(x+1)}$$ where $P(x)$ is a polynomial. 3. **Horizontal asymptote at $y=0$:** This implies the degree of $P(x)$ is less than the degree of the denominator, so $\deg(P) < 2$. 4. **Finding $x$-intercepts:** The $x$-intercepts occur where $f(x) = 0$, which means the numerator $P(x) = 0$. 5. **Given the graph behavior:** The branch between $x = -2$ and $x = -1$ crosses downward through the center, indicating an $x$-intercept in that interval. 6. **Assuming $P(x)$ is linear:** Let $P(x) = a(x - r)$ where $r$ is the $x$-intercept. 7. **Since the $x$-intercept is between $-2$ and $-1$, $r \in (-2, -1)$. 8. **Conclusion:** The graph has exactly one $x$-intercept located between $x = -2$ and $x = -1$. **Final answer:** The $x$-intercept is a single point with $x$-coordinate between $-2$ and $-1$. No other $x$-intercepts exist based on the graph description.