Is $f(x) = 3x^4 - 2x^2 + 5$ even, odd or neither?

Even: its graph is symmetric about the $y$-axis.

Is $g(x) = x^3 - 4x$ even, odd or neither?

Odd: its graph has point symmetry about the origin.

Show that $h(x) = x^3 + x^2 - 1$ is neither even nor odd.

Neither: the powers are mixed, so the graph has neither symmetry.

Is $p(x) = (x - 1)(x + 1)(x^2 + 3)$ even, odd or neither?

Even, even though the zeros $\pm 1$ are the only hint in the factored form.

For $f(x) = 2x^5 - kx^3 + (k - 4)x^2 + 6x$, find every value of $k$ that makes $f$ odd, and explain why no value of $k$ makes it even.

$f$ is odd only when $k = 4$, and is never even.

Grade 12 · MHF4U · Advanced Functions

MHF4U: Even and Odd Functions

An even function is symmetric about the $y$ -axis; an odd function is symmetric about the origin. Both are tests you can run on the equation in one line, by replacing $x$ with $-x$ .

How It Works

Even: $f(-x) = f(x)$ for every $x$ . Fold the graph along the $y$ -axis and the halves match. $y = x^2$ is the model: $(-3)^2 = 3^2$ .

Odd: $f(-x) = -f(x)$ for every $x$ . Rotate the graph half a turn about the origin and it lands on itself. $y = x^3$ is the model: $(-2)^3 = -8 = -(2^3)$ . An odd function that is defined at $0$ must pass through the origin, since $f(0) = -f(0)$ forces $f(0) = 0$ .

For a polynomial the test is just the exponents. Only even powers, constant included since $5 = 5x^0$ : even. Only odd powers: odd. Any mix: neither. That is because $(-x)^n$ equals $x^n$ when $n$ is even and $-x^n$ when $n$ is odd, so a mixed polynomial keeps some signs and flips others.

To show it algebraically, write $f(-x)$ , simplify each term, and compare with $f(x)$ and with $-f(x)$ . Do both comparisons: a function that fails the even test is not automatically odd. Most functions are neither. A quick numeric check helps too: compare $f(2)$ with $f(-2)$ . Equal points to even, negatives of each other points to odd, and anything else settles it as neither.

A factored polynomial hides its powers, so expand first or test $f(-x)$ directly. And watch the names: an even function and an even degree are different things. $x^4 + x$ has even degree and is neither.

5 Worked Examples

Easy first, test level last. Try each one on paper, then open the solution.

Easy
Is $f(x) = 3x^4 - 2x^2 + 5$ even, odd or neither?
Show solution
1. $f(-x) = 3(-x)^4 - 2(-x)^2 + 5 = 3x^4 - 2x^2 + 5 = f(x)$ .
2. Every power is even (the constant is $5x^0$ ), which says the same thing.
Answer: Even: its graph is symmetric about the $y$ -axis.
Easy
Is $g(x) = x^3 - 4x$ even, odd or neither?
Show solution
1. $g(-x) = (-x)^3 - 4(-x) = -x^3 + 4x$ .
2. That is $-(x^3 - 4x) = -g(x)$ , and both powers are odd.
Answer: Odd: its graph has point symmetry about the origin.
Medium
Show that $h(x) = x^3 + x^2 - 1$ is neither even nor odd.
Show solution
1. $h(-x) = -x^3 + x^2 - 1$ .
2. Compare with $h(x) = x^3 + x^2 - 1$ : the $x^3$ term flipped sign, so $h(-x) \ne h(x)$ . Not even.
3. Compare with $-h(x) = -x^3 - x^2 + 1$ : the $x^2$ and constant terms differ, so $h(-x) \ne -h(x)$ . Not odd.
4. A number check agrees: $h(2) = 11$ but $h(-2) = -5$ , which is neither $11$ nor $-11$ .
Answer: Neither: the powers are mixed, so the graph has neither symmetry.
Medium
Is $p(x) = (x - 1)(x + 1)(x^2 + 3)$ even, odd or neither?
Show solution
1. The factored form hides the powers, so expand: $(x - 1)(x + 1) = x^2 - 1$ , then $(x^2 - 1)(x^2 + 3) = x^4 + 2x^2 - 3$ .
2. Only even powers, so $p(-x) = p(x)$ .
Answer: Even, even though the zeros $\pm 1$ are the only hint in the factored form.
Test level
For $f(x) = 2x^5 - kx^3 + (k - 4)x^2 + 6x$ , find every value of $k$ that makes $f$ odd, and explain why no value of $k$ makes it even.
Show solution
1. Odd needs odd powers only. The $x^5$ , $x^3$ and $x$ terms are fine; the $x^2$ term must vanish: $k - 4 = 0$ , so $k = 4$ .
2. Check: with $k = 4$ , $f(x) = 2x^5 - 4x^3 + 6x$ and $f(-x) = -2x^5 + 4x^3 - 6x = -f(x)$ .
3. Even needs even powers only, but the $2x^5$ term does not contain $k$ and never disappears, so $f(-x) \ne f(x)$ for every $k$ .
Answer: $f$ is odd only when $k = 4$ , and is never even.

Learn It on Martian Lab

Martian Lab teaches this in the Symmetry and Even & Odd Functions lesson of its MHF4U Polynomial Functions unit. Watch it, then take the Polynomial Functions practice tests to see if it stuck.

Watch the Symmetry and Even & Odd Functions lesson free

Keep Going

Free Polynomial Functions practice test, with solutions

More Polynomial Functions topics: End Behaviour and Degree · Finding the Equation of a Polynomial

More MHF4U topics: Long and Synthetic Division · Vertical and Horizontal Asymptotes · Proving Trig Identities · Solving Trig Equations · Laws of Logarithms · Solving Logarithmic Equations · Remainder and Factor Theorem · Solving Polynomial Equations · Polynomial Inequalities · Sum and Difference of Cubes · Holes in Rational Functions · Graphing Rational Functions · Rational Equations and Inequalities · Radian Measure · Unit Circle and Special Angles · Compound Angle Formulas · Double Angle Formulas · Graphing Sine and Cosine with Transformations · Graphs of Tan, Sec, Csc and Cot · Sinusoidal Word Problems · Solving Exponential Equations · Graphing Logarithmic Functions · Exponential and Log Word Problems · Sums, Products and Quotients of Functions · Composition of Functions · Average vs Instantaneous Rate of Change

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