Determine End Behavior of

Question

Determine the end behavior of

f(x)= = -12x^4 + 23x³ - 5x^2 - 23x +9 / -4x^3 + 5x^2 +2x - 4

Answer 1

To determine the end behavior of the function, we need to analyze what happens to the function as x approaches positive infinity and negative infinity.

We can start by looking at the degree of the numerator and denominator of the function:

1. The degree of the numerator is 4
2. The degree of the denominator is 3

Since the degree of the numerator is greater than the degree of the denominator, we know that as x approaches positive or negative infinity, the function will approach either positive or negative infinity.

To determine the sign of the function at infinity, we can look at the leading coefficient of the numerator and denominator:

1. The leading coefficient of the numerator is -12
2. The leading coefficient of the denominator is -4

Since the leading coefficient of the numerator and denominator are both negative, we can conclude that as x approaches positive infinity, the function will approach positive infinity and as x approaches negative infinity, the function will approach negative infinity.

Therefore, the end behavior of the function is:

1. As x approaches positive infinity, f(x) approaches positive infinity
2. As x approaches negative infinity, f(x) approaches negative infinity.