Question

Determine whether the given function is even, odd, or neither.

f(x) = x^4 – x^3
A) Neither
B) Even
C) Odd

f(x) = x^3 - 2x
A) Neither
B) Even
C) Odd

f(x) = 2x^2 + x^4
A) Even
B) Neither
C) Odd

Answer 1

The first function is even, the second function is odd, and the third function is even.

Step-by-step explanation:

An even function is symmetric about the y-axis, which means that its graph is unchanged when reflected across the y-axis. To determine if a function is even, we substitute -x for x in the function and check if it remains the same. If it does, the function is even. If it changes sign, the function is odd. If neither of these conditions are met, the function is neither even nor odd.

For the given functions:

f(x) = x⁴ – x³

Replacing x with -x, we get f(-x) = (-x)⁴ - (-x)³ = x⁴ + x³

Since f(-x) = f(x), the function is even.

f(x) = x³ - 2x

Replacing x with -x, we get f(-x) = (-x)³ - 2(-x) = -x³ + 2x

Since f(-x) = -f(x), the function is odd.

f(x) = 2x² + x⁴

Replacing x with -x, we get f(-x) = 2(-x)² + (-x)⁴ = 2x² + x⁴

Since f(-x) = f(x), the function is even.