Question

The rational function age is given by age of x equals 2x⁵ + 5x³ - 2x² - 13/3x² - 2x + 7. Which of the following describes the end behavior of h?

1) The end behavior of h is increasing as x approaches positive infinity and decreasing as x approaches negative infinity.
2) The end behavior of h is decreasing as x approaches positive infinity and increasing as x approaches negative infinity.
3) The end behavior of h is increasing as x approaches both positive and negative infinity.
4) The end behavior of h is decreasing as x approaches both positive and negative infinity.

Answer 1

The end behavior of the given rational function is increasing as x approaches positive infinity and decreasing as x approaches negative infinity.

Step-by-step explanation:

The given rational function is h(x) = (2x⁵ + 5x³ - 2x² - 13)/(3x² - 2x + 7). To determine the end behavior of h, we need to examine the highest degree terms in the numerator and denominator.

For large positive values of x, the highest degree term in the numerator is 2x⁵ and the highest degree term in the denominator is 3x². Since the degree of the numerator is higher than the degree of the denominator, as x approaches positive infinity, the value of h(x) will also approach positive infinity. Therefore, the end behavior of h is increasing as x approaches positive infinity.

Similarly, for large negative values of x, the highest degree term in the numerator is 2x⁵ and the highest degree term in the denominator is 3x². Again, the degree of the numerator is higher than the degree of the denominator, so as x approaches negative infinity, the value of h(x) will also approach negative infinity. Therefore, the end behavior of h is decreasing as x approaches negative infinity.

Therefore, the correct description of the end behavior of h is: The end behavior of h is increasing as x approaches positive infinity and decreasing as x approaches negative infinity.