Foundations — Even and odd functions — graphical and algebraic tests

Before we can ask that question honestly, we have to earn every symbol the parent note tossed around: $x$, $-x$, $f(x)$, $f(-x)$, the graph, the y-axis, the origin, "domain", and the minus sign out front. Let's build them one at a time, from nothing.

1. A number line, and what $x$ means

Picture a ruler stretching left and right forever. The middle mark is $0$. To the right the marks grow: $1,2,3,\dots$. To the left they go negative: $-1,-2,-3,\dots$. This ruler is called the number line.

Look at the red dot at $x=2$. It sits two steps right of $0$. Everything in this topic starts from picking such a spot.

2. Flipping the sign: what $-x$ does

The minus sign here is not "subtract". It is a flip: it grabs your spot and swings it across $0$ to the mirror position. In the figure above, the blue dot at $-2$ is exactly the red dot at $2$ flipped over the centre.

Careful, and this bites people: if $x=-3$, then $-x=-(-3)=+3$. Two flips cancel. The minus sign flips whatever is there, sign included.

3. The function machine $f(x)$

The bracket is not multiplication. $f(x)$ does not mean $f$ times $x$. It means "feed $x$ into the machine $f$ and read what comes out".

In the figure, the number $x$ walks into the box labelled $f$, and a (possibly different) number $f(x)$ walks out. For the rule $f(x)=x^2+4$, feeding in $3$ gives out $3^2+4=13$, so $f(3)=13$.

4. $f(-x)$ — the star of the show

This combines Section 2 (the flip) with Section 3 (the machine). Order matters: flip first, feed second.

The parent note's two definitions are now readable:

• Even: $f(-x)=f(x)$ — flipping the input changes nothing in the output.
• Odd: $f(-x)=-f(x)$ — flipping the input flips the output's sign too.

5. Drawing it: axes, points, and the graph

The crossing point $(0,0)$ is called the origin. The graph of $f$ is the collection of all dots $(x,f(x))$ as $x$ sweeps along the whole number line — a curve tracing every input-output pair at once.

The figure shows why the algebra turns into a picture:

• Even (left, blue $x^2$): the point $(a,b)$ and its mirror $(-a,b)$ sit at the same height. Fold along the yellow y-axis and the halves land on each other. That fold-symmetry is $f(-x)=f(x)$.
• Odd (right, green $x^3$): the point $(a,b)$ pairs with $(-a,-b)$ — flipped left-right and up-down. That is a 180° spin about the origin, which is $f(-x)=-f(x)$.

6. The domain — the machine's legal inputs

Some machines refuse certain inputs. $\sqrt{x}$ chokes on negatives, so its domain is $x\ge 0$. This matters enormously here: to compare $f(x)$ with $f(-x)$, both $x$ and $-x$ must be legal inputs. A domain must be symmetric about $0$ before the even/odd question even makes sense — see Domain and range.

In the figure the green region ($x\ge 0$ only) is lopsided: input $2$ is allowed but its mirror $-2$ is banned, so we can never compare — such a function is automatically neither. The blue region (all of $\mathbb{R}$) is balanced, so testing is allowed.

7. The minus-out-front, and factoring it

The odd test hinges on spotting $-f(x)$ hidden inside $f(-x)$. That is an algebra skill: factoring out a negative.

Prerequisite map

Each box is a symbol we built above; the arrows show which ideas must exist before the even/odd tests at the bottom make sense. The parent note lives at that bottom node — see the topic note.

Where these foundations lead

• The mirror pictures feed straight into Function transformations (reflections across axes).
• The evenness of $\cos$ and oddness of $\sin$ come from Trigonometric functions.
• The even/odd parts of $e^x$ define Hyperbolic functions.
• Symmetry that makes integrals vanish is used all over Symmetry in calculus and Fourier series.

Equipment checklist

Cover the right side and see if you can answer before revealing.