Given the function \( f(x) = -2x^3 + 3x^2 - 2x + 1 \), determine whether it is symmetrical about the y-axis (even), symmetrical about the origin (odd), or neither.
Answer
Next, we check if the function is odd. A function is odd if it satisfies the condition \( f(-x) = -f(x) \). We compare \( f(-x) \) and \( -f(x) \): \( -f(x) = 2x^3 - 3x^2 + 2x - 1 \). This is also not equal to \( f(-x) \), so the function is not odd.
Step 1 ：First, we check if the function is even. A function is even if it satisfies the condition \( f(-x) = f(x) \). We compute \( f(-x) \): \( -2(-x)^3 + 3(-x)^2 - 2(-x) + 1 = -2x^3 + 3x^2 + 2x + 1 \). This is not equal to \( f(x) \), so the function is not even.
Step 2 ：Next, we check if the function is odd. A function is odd if it satisfies the condition \( f(-x) = -f(x) \). We compare \( f(-x) \) and \( -f(x) \): \( -f(x) = 2x^3 - 3x^2 + 2x - 1 \). This is also not equal to \( f(-x) \), so the function is not odd.
