Question

Consider the following rational function \[f\]. \[f(x)=\dfrac{6x^3-x^2 7}{2x 5}\] determine \[f\]'s end behavior. \[f(x)\to\] as \[x\to -\infty\]. \[f(x)\to\] as \[x\to \infty\]

Answer 1

The end behavior of the rational function f(x) shows that as x approaches infinity, f(x) approaches infinity, and as x approaches negative infinity, f(x) approaches negative infinity due to the dominant x^3 term.

Step-by-step explanation:

To determine the end behavior of the rational function f(x), we examine the behavior as x approaches negative infinity (x → -∞) and positive infinity (x → ∞). As x gets very large (positively or negatively), the highest power terms in the numerator and denominator dictate the function's growth. For the function f(x) = ⁡() = () = ∕, the leading term in the numerator is 6x^3 and in the denominator it's 2x. When x → ∞, the x^3 term will dominate, meaning f(x) will go to infinity after simplifying to 3x^2. When x → -∞, the x^3 term will still dominate and will result in negative infinity since a negative value raised to an odd power is negative. Therefore, f(x) → ∞ as x → ∞ and f(x) → -∞ as x → -∞.