# 9. Even and Odd Functions

By M. Bourne

## Even Functions

A function `y = f(t)` is said to be **even** if

f(−t) =f(t)

for all values of *t*.

The graph of an **even** function is always symmetrical about the **vertical axis** (that is, we have a mirror image through the *y*-axis).

The waveforms shown below represent **even** functions:

### Cosine curve

*f*(*t*) = 2 cos *πt*

Graph of *f*(*t*) = 2 cos(π*t*), an even function.

Notice that we have a mirror image through the `f(t)` axis.

### Even Square wave

Graph of an even step function.

### Triangular wave

Graph of an even triangular function.

In each case, we have a mirror image through the `f(t)` axis. Another way of saying this is that we have **symmetry** about the vertical axis.

## Odd Functions

A function `y=f(t)` is said to be **odd** if

`f(-t) = - f(t)`

for all values of *t*.

The graph of an **odd** function is always symmetrical about the **origin**.

### Origin Symmetry

A graph has **origin symmetry** if we can fold it along the **vertical** axis, then along the **horizontal** axis, and it lays the graph onto itself.

Another way of thinking about this is that the graph does exaclty the **opposite** thing on each side of the origin.
If the graph is going up to the right on one side of the origin, then it will be going down to the left by the same amount on the other side of the origin.

### Examples of Odd Functions

The waveforms shown below represent **odd** functions.

### Sine Curve

*y*(*x*) = sin *x*

Graph of *y*(*x*) = sin(*x*), an odd function.

Notice that if we fold the curve along the *y*-axis, then along the *t*-axis, the graph maps onto itself. It has **origin symmetry**.

### "Saw tooth" wave

Graph of a sawtooth function which is odd.

### Odd Square wave

Graph of an odd square wave.

Each of these three curves is an **odd function**, and the graph demonstrates **symmetry about the origin**.

## Exercises

### Need Graph Paper?

Sketch each function and then determine whether each function is odd or even:

(a) `f(t)={(e^t,text(if ) -pi le t lt 0),(e^-t,text(if ) 0 le t lt pi):}`

Answer

Graph of a split function.

We can see from the graph that it is
**even**.

**OR:** The function is **even**
since `f(−t) = f(t)` for all
values of *t*.

(b) `f(t)={(-1,text(if ) 0 le t lt pi/2),(1,text(if ) pi/2 le t lt (3pi)/2),(-1,text(if ) (3pi)/2 le t lt 2pi) :}`

and

f(t) =f(t+ 2π)

(This last line means: Periodic with period = 2π)

Answer

Graph of a step function.

We can see from the graph that it is
**even**.

**OR:** The function is **even**
since `f(−t) = f(t)` for all
values of *t*.

(c) `f(t)={(-t+pi,text(if ) -pi le t lt 0),(-t-pi,text(if ) 0 le t lt pi):}`

Answer

Graph of a split function.

We can see from the graph that the function is
**odd**.

**OR:** The function is **odd** since `f(−t) = -f(t)` for
all values of *t*.

(d) `f(t)={(t-pi,text(if ) -pi le t lt 0),(-t+pi,text(if ) 0 le t lt pi):}`

Answer

Graph of a split function.

We can see from the graph that it is **neither**
odd nor even.

(e) `f(t)={(t+pi,text(if ) -pi le t lt 0),(-t+pi,text(if ) 0 le t lt pi):}`

Answer

Graph of a split function.

We can see from the graph that it is
**even**.

**OR:** The function is **even**
since `f(−t) = f(t)` for all
values of *t*.

(f) `f(t)={((t+pi/2)^2,text(if ) -pi le t lt 0),(-(t-pi/2)^2,text(if ) 0 le t lt pi):}`

Answer

Graph of a split function.

We can see from the graph that the function is
**odd**.

**OR:** The function is **odd** since `f(−t) = -f(t)` for
all values of *t*.