Final answer:
To find the even and odd parts of the function f(x) = 2x² - 5x + 15, substitute -x for x in the function and check if it remains the same. Since f(x) = f(-x), the function is even. The even part is 2x² + 5x + 15, and the odd part is -10x.
Step-by-step explanation:
An even function is a function that satisfies the equation y(x) = y(-x), while an odd function is a function that satisfies the equation y(x) = -y(-x). To find the even and odd parts of the function f(x) = 2x² - 5x + 15, we need to determine whether the function is even or odd.
To determine if a function is even, we substitute -x for x and check if the function remains the same. Let's substitute -x for x in the given function:
f(-x) = 2(-x)² - 5(-x) + 15 = 2x² + 5x + 15
Since f(x) = f(-x) for the given function, it is an even function. The even part of the function is 2x² + 5x + 15 because it remains the same after substituting -x for x.
The odd part of the function is obtained by subtracting the even part from the original function: odd part = f(x) - even part = (2x² - 5x + 15) - (2x² + 5x + 15) = -10x.