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(πt), an even function.

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

Continues below

Even Square wave

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

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

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

Exercises

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.

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(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.

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

Answer

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

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

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

Answer

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.

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(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.