If the target of one transformation ` S`:

Example The composition

(`ID _{p}`)(

is the same as

`ID Number of the Instructor`: course → number

The composition

(`Population`)(`Capital City`): country → city → number

is the same as

`Population of the Capital City`: country → number

Example We are familiar with the following exercise:
Suppose `y` = `x` + 1 and `z` = `y`^{2}.
Then `z` = (`x` + 1)^{2} = `x`^{2} + 2`x` + 1.
What we have done is actually composing two transformations

`Add One`: **R** → **R**, `x` → `x` + 1

`Square`: **R** → **R**, `y` → `y`^{2}

to form a transformation

`Add One and then Take square`: **R** → **R**, `x` → (`x` + 1)^{2}

For a multi-variable example, consider the transformations

` T`:

The composition ** TS** is computed by substituting the formulae of

`z`_{1}
= (`x`_{1} + `x`_{2})^{2} - 2`x`_{1}`x`_{2}
= `x`_{1}^{2} + `x`_{2}^{2}

`z`_{2}
= (`x`_{1} + `x`_{2})^{3} - 3(`x`_{1} + `x`_{2})`x`_{1}`x`_{2}
= `x`_{1}^{3} + `x`_{2}^{3}

`z`_{3}
= (`x`_{1} + `x`_{2})^{4} - 4(`x`_{1} + `x`_{2})^{2}`x`_{1}`x`_{2} - 2(`x`_{1}`x`_{2})^{2}
= `x`_{1}^{4} + `x`_{2}^{4}

Thus we have

` TS`(

If transformations are given by formulae, then the composition is simply the substitution.

Example
Let ` T` and

From the picture, we see that the composition ** TS** is the reflection with respect
to the line