Odd and Even Functions

Try this

For $f (x) = x^{3} - 3 x$ , press “Fold on the y-axis”: does the dashed copy land on the original graph? What about “Half-turn about O”?
Drag the probe $x_{0}$ . Point $A$ sits at $x_{0}$ and point $B$ at $- x_{0}$ . For an odd function, how are the lengths and directions of the two vertical segments related?
Open “Change coefficients and domain” and keep only $b$ (the $x^{2}$ term) and $d$ (the constant). Which light is on now? What if you keep only $a$ and $c$ ?
Change the left end of the domain from $- 3$ to $- 2$ . The formula has not changed, so why did both lights go out?

Let the domain $D$ of $f (x)$ be symmetric about the origin. If $f (- x) = f (x)$ for every $x \in D$ , then $f (x)$ is an even function; if $f (- x) = - f (x)$ for every $x \in D$ , it is an odd function.

even ⟺ the graph is symmetric about the y -axis, odd ⟺ the graph is symmetric about the origin .

For polynomials only the degrees matter: with only even-degree terms (constants included) the function is even; with only odd-degree terms it is odd.

Predict first

$f (x)$ is an odd function on $R$ , and $f (x) = x^{2} - 2 x$ for $x > 0$ . Find $f (- 1)$ .

Choose an answer first; then you can see the result.

Why: every function splits into an even part and an odd part

As long as the domain is symmetric about the origin,

f (x) = g (x) \frac{f ( x ) + f ( - x )}{2} + h (x) \frac{f ( x ) - f ( - x )}{2} .

Example: write f(x) = x³ + x² − 2x + 1 as an even function plus an odd function

Step 1 / 5

It is neither odd nor even
Fold it: the dashed copy $f (- x)$ does not match the graph. Turn it half a turn: $- f (- x)$ does not match either. Both lights show ✗.

$f (- x) = - x^{3} + x^{2} + 2 x + 1$
Average to make an even function
Half the difference makes an odd function
Check at one point
The split is unique

Remove one condition

Break it

Counterexample lab: remove one condition

Conclusion: $f (x) = x^{2}$ is an even function

The domain is $[- 3, 3]$ . For every $x_{0}$ , $A$ and $B$ are at the same height, and the folded copy matches the graph exactly.

Practice

Exercise 1Basic

The function $f (x) = (x + a) (x - 4)$ is even. Find $a$ .

Exercise 2Basic

$f (x)$ is an odd function on $R$ , and $f (x) = x^{2} + x$ for $x > 0$ . Find $f (- 2)$ .

Exercise 3Advanced

$f (x) = a x^{3} + b x + 2$ , and $f (- 3) = 5$ . Find $f (3)$ .

Exercise 4Challenge

Write $f (x) = 2^{x}$ as the sum of an even function $g (x)$ and an odd function $h (x)$ (both with domain $R$ ). Find $g (1)$ .

Try this

Why: every function splits into an even part and an odd part

Example: write f(x) = x³ + x² − 2x + 1 as an even function plus an odd function

Remove one condition

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

The graph of an odd function always passes through the origin

Simplifying first, then deciding odd or even