Example 1: an even polynomial

• Show that f(x) = x⁴ − 3x² + 1 is even.
• Solution f(−x) = (−x)⁴ − 3(−x)² + 1 = x⁴ − 3x² + 1 = f(x). So f is even.

Example 2: an odd polynomial

• Show that h(x) = x³ − 5x is odd.
• Solution h(−x) = (−x)³ − 5(−x) = −x³ + 5x = −(x³ − 5x) = −h(x). So h is odd.

Example 3: a rational function

• Study the parity of g(x) = x / (x² + 1).
• Solution The denominator x² + 1 is always positive, so the domain is all real numbers, which is symmetric. g(−x) = −x / ((−x)² + 1) = −x / (x² + 1) = −g(x). So g is odd.

Example 4: a function that is neither because of its domain

• Study the parity of q(x) = √x.
• Solution The domain is [0, ∞[. The number 2 belongs to the domain but −2 does not, so the domain is not symmetric about zero. Therefore q cannot be even or odd.

Example 5: using symmetry to find points

• The function f is even, and the points (2, 5) and (3, −1) lie on its graph. Which other points are known on the graph?
• Solution Because f is even, f(−2) = f(2) = 5 and f(−3) = f(3) = −1. So (−2, 5) and (−3, −1) also lie on the graph.

Example 6: finding a constant that makes a function odd

• Let f(x) = x³ + ax + b. Find the value of b that makes f odd.
• Solution f(−x) = −x³ − ax + b. For f to be odd, this must equal −f(x) = −x³ − ax − b. Comparing the constant terms gives b = −b, so 2b = 0 and b = 0. The value of a does not change this result, so any value of a gives an odd function.