Find the end behavior of the rational function: \( f(x)=\frac{x^{2}-x-12}{x+1} \)

To analyze the end behavior of the rational function \( f(x) = \frac{x^2 - x - 12}{x + 1} \), we can look at the degrees of the polynomial in the numerator and the polynomial in the denominator. 1. **Identify the degrees**: - The degree of the numerator \( x^2 - x - 12 \) is 2. - The degree of the denominator \( x + 1 \) is 1. 2. **Determine the end behavior**: Since the degree of the numerator (2) is greater than the degree of the denominator (1), the end behavior of the function can be determined by the leading terms of the numerator and denominator. The leading term of the numerator is \( x^2 \) and the leading term of the denominator is \( x \). Therefore, we can approximate the function for large values of \( |x| \): \[ f(x) \approx \frac{x^2}{x} = x \] 3. **Conclusion**: As \( x \) approaches positive or negative infinity, \( f(x) \) behaves like \( x \). Thus, we can summarize the end behavior as follows: - As \( x \to \infty \), \( f(x) \to \infty \). - As \( x \to -\infty \), \( f(x) \to -\infty \). In conclusion, the end behavior of the function \( f(x) = \frac{x^2 - x - 12}{x + 1} \) is: - \( f(x) \to \infty \) as \( x \to \infty \) - \( f(x) \to -\infty \) as \( x \to -\infty \)