In this free pre calculus help section we will look at the following types of functions:

- Identity Function
- Even and Odd Function

We will also look at examples to differentiate the types of the Even and Odd Functions. We can learn to find the differences between Even and Odd Functions.

## Identity Function

Identity function is defined as a function *f *from a set A to the same set A such that *f*(*x*) = *x *for all *x *in A.

So we have *f:* A $\rightarrow$ A defined by *f*(*x*) = *x *for all *x *in A.

Identity function is usually denoted by I_{A} or simply I.

Hence I(*x*) = *x *always.

The identity function on a set *A* can be understood as a function that *does nothing *to each element of *A.*

## Even Function

Even and Odd Functions are also a tye of functions.

An even function is defined as a function *f *such that ** f(x) = f (-x)** for all

*x*in the domain.

In other words, we say that a function is even if replacing *x* with -*x* does NOT change the original function.

Hence if you are asked to determine algebraically whether a function is even, all that you have to do is replace x by –x. If you still end up with the original function then it is an even function

Solved pre calculus problem :-

Consider f(x) = x^{2}

Replacing x by –x we get, f (-x) = (-x)^{2} = x^{2} which is same as the original f(x).

Hence it is an even function.

Also for g(x) = |x| we get g (-x) = |-x| = |x| = g(x)

Hence the modulus function is also an even function.

Graphically, an even function is symmetric with respect to y-axis. In other words, even functions are those for which the left half of the plane looks like the mirror image of the right half of the plane.

The following examples will make this point more clear.

## Odd Function

Even and Odd Functions include the study of Odd functions too.

An odd function is defined as a function *f *such that ** f (-x) = -f ( x)** for all

*x*in the domain.

In other words, we say that a function is odd if replacing *x* with -*x* results in change in all the signs in the original function.

Hence if you are asked to determine algebraically whether a function is Even and Odd Functions, all that you have to do is replace x by –x. If you end up with the negative of the original function then it is an odd function.

For example,

Consider f(x) = x^{3}

Replacing x by –x we get, f (-x) = (-x)^{3} = -x^{3} = -f(x).

Hence it is an odd function.

Graphically, an odd function is symmetric with respect to the origin (i.e. if you draw a line from one point on the graph, say f(x) passing through the origin extending the same length across the origin you will reach f (-x)) .In other words, these are functions where the left half of the plane looks like the inverted mirror image of the right half of the plane.

Following free precalculus answers Even and Odd Functions examples give us the picture of odd functions.