Final Answer:
The statement that BEST describes the end behavior of f(x) = 9 - 3x - 4x⁴ + 2x⁵ is B) f(x) approaches [infinity] as x approaches [infinity], and f(x) approaches -[infinity] as x approaches -[infinity].
Step-by-step explanation:
To determine the end behavior of a polynomial function, we look at the term of the polynomial with the highest power, known as the leading term. The coefficients of the other terms do not affect the end behavior.
The function in question is f(x) = 9 - 3x - 4x⁴ + 2x⁵
The leading term here is 2x⁵, as it has the highest power of x. The coefficient of the leading term is the number in front of the x⁵, which is 2, a positive number. The highest power, which is the degree of the polynomial, is 5, an odd number.
Now, because the coefficient of the leading term is positive, as x approaches positive infinity, 2x⁵ will also approach positive infinity. As x gets larger and larger, the positive power of x to the fifth power will overwhelm all the other terms in the polynomial, so they become insignificant in determining the behavior of f(x) at very large values of x.
Similarly, as x approaches negative infinity, since the leading term has an odd power, 2x⁵ will approach negative infinity. An odd power preserves the sign of x, therefore, with a negative x raised to the fifth power, the result will be negative.
Given these observations, the end behavior of the polynomial function can be described as follows:
- f(x) approaches positive infinity as x approaches positive infinity because the leading coefficient is positive and the leading term has an odd degree.
- f(x) approaches negative infinity as x approaches negative infinity because an odd power retains the negative sign when x is negative.
Therefore, the statement that BEST describes the end behavior of the function f(x) = 9 - 3x - 4x⁴ + 2x⁵ is:
B) f(x) approaches [infinity] as x approaches [infinity], and f(x) approaches -[infinity] as x approaches -[infinity].