Final answer:
Functions f(x) and h(x) have the same end behaviors as they are both even degree polynomials with a positive leading coefficient, meaning both ends point up. The function g(x) is an odd degree polynomial, which means it has opposite end behaviors, where one side of the graph tends towards positive infinity and the other towards negative infinity.
Step-by-step explanation:
The question is about determining which functions have opposite end behaviors. To analyze the end behaviors of functions, we should look at the highest degree term of each function because it dictates the end behavior as x approaches infinity or negative infinity.
Function f(x) = x² + 3x - 1 is a quadratic function where the leading term is x². Since the highest power is even, both ends of the graph will either point up or down, depending on the sign of the leading coefficient, which is positive in this case, so both ends point up.
Function g(x) = x³ + x² + 3x - 1 is a cubic function with the leading term x³. This is an odd degree function, so its end behaviors are opposite; as x approaches negative infinity, f(x) approaches positive infinity, and as x approaches positive infinity, f(x) approaches negative infinity.
Lastly, function h(x) = x⁴ + 3/3x² + 3x - 1 is a quartic function with the leading term x⁴. This function, like f(x), will have the same end behavior on both sides because it is an even degree function with a positive leading coefficient, meaning both ends point up.
Therefore, the functions with opposite end behaviors are f(x) and h(x), as both are quadratic functions with leading terms that are positive and even degrees, meaning both ends point up, unlike g(x), which has an odd degree and opposite end behaviors.