Example 1: an even polynomial

• Show that f(x) = x⁴ − 2x² is even.
• Solution f(−x) = (−x)⁴ − 2(−x)² = x⁴ − 2x² = f(x). So the function is even.

Example 2: an odd polynomial

• Show that g(x) = x³ − x is odd.
• Solution g(−x) = (−x)³ − (−x) = −x³ + x = −(x³ − x) = −g(x). So the function is odd.

Example 3: a function that is neither even nor odd

• Study the parity of h(x) = x² + x.
• Solution h(−x) = x² − x. This is not equal to h(x) = x² + x, unless x = 0. Also h(−x) is not equal to −h(x) = −x² − x. For example, h(1) = 2 while h(−1) = 0. So h is neither even nor odd.

Example 4: finding a value using evenness

• The function f is even and f(5) = −3. Find f(−5).

Example 5: finding values using oddness

• The function q is odd and q(2) = 7. Find q(−2), then find q(0) if the function is defined at zero.
• Solution q(−2) = −q(2) = −7. For an odd function defined at 0 we have q(0) = −q(0), so 2q(0) = 0 and q(0) = 0.

Example 6: proving a function is one-to-one

• Show that f(x) = 5x − 1 is one-to-one.
• Solution Suppose f(a) = f(b). Then 5a − 1 = 5b − 1, so 5a = 5b and a = b. Therefore the function is one-to-one.