Final answer:
The end behavior of the function h(x) = 2x⁶ - 2x² + 2 as x approaches positive infinity is that it approaches positive infinity, dominated by its leading term 2x⁶.
Step-by-step explanation:
The question asks about the end behavior of the polynomial function h(x) = 2x⁶ - 2x² + 2 as x approaches positive infinity. To determine the end behavior, we can focus on the term with the highest power of x, as it will dominate the function's growth for large values of x. Thus, the most significant term is 2x⁶ because as x approaches positive infinity, the contributions from the -2x² and +2 terms become negligible in comparison to 2x⁶.
As a result, the end behavior of h(x) is similar to the end behavior of 2x⁶, which is that of a positive, even-degree polynomial. Consequently, the function h(x) will also approach positive infinity as x approaches positive infinity. This is because the leading coefficient 2 is positive, and the highest power 6 is even, ensuring that the output grows without bound in the positive direction as x becomes larger and larger.