Composition

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We have studied the concept of functions and how we can express them in the form of equations. But have you ever wondered, what if two functions are given together and their combination forms an entirely new function? And if such a combination also occurs when handling shapes using Geometry, then how can we deal with it? These kinds of complicated functions and shape transformations can be solved using the concept of composition. In this section, we will understand what composition is and how it is applied and used.

Meaning of composition

In mathematics, we can generate a singular, more complex function from multiple simple functions. This process is known as composition, where we consider the output of one function as the input of another function.

A composition of functions is a mathematical process where two functions are combined to generate a composite function. The output of one function is used as the input of another.

We can denote composite functions as $f \circ g$ or $T \circ S$ , where f and g are functions and T and S are transformations. The composition can also be denoted as $f (g (x)) .$ The composition of two functions f and g gives a new function h. It is important to know the functions' order, as this affects the production of the new composite function. For example, using the output of function f as the input of function g to obtain composite function h is not the same as using function g as the input of function f.

We may also use the concept of composition in Geometry, called the composition of transformations, which occurs on shapes (or figures) rather than functions. Transformations in Geometry include translations, reflections, rotations, etc.

A composition of transformations is a process where a combination of transformations is performed on a shape or figure consecutively, with the resulting transformed shape from one transformation used as the starting point in the following transformation.

In this article we explore composition both in terms of transformations and functions.

Composition of transformations

Let us consider a transformation T which maps from point A to B as well as another transformation S which maps from point B to C. From these transformations T and S, we can produce a new composite transformation that can map directly from point A to C, called the composite of S and T. We denote it as $S \circ T$ and state it as "S of T."

Composition, composition of transformation, StudySmarter Composition of transformations, StudySmarter Originals

Here, we apply transformation T first, and then we apply transformation S. So, we can represent this composition as: $C = S (B) = S (T (A)) = (S \circ T) (A)$

An example of a composition of transformations is a glide reflection. It is the composition of a reflection and a translation.

Statements regarding composition of transformations

Let's take a look at some of the theorems concerning the composition of transformations.

Theorem 1: Composition Theorem

A composition of two isometries is an isometry.

Theorem 2: Reflections in Parallel Lines Theorem

A composition of reflections in two parallel lines is a translation. That is, for two parallel lines l and m, the reflection in l and then the reflection in m is the same as a translation.

Theorem 3: Reflection in Intersecting Lines Theorem

A composition of reflections in two intersecting lines is a rotation about the point of intersection of those two lines. In other words, if l and m are two lines intersecting at point O, then reflection in l and then the reflection in m is the same as a rotation about point O.

Theorem 4: Reflection in Perpendicular Lines

A composition of reflections in perpendicular lines is a half-turn about the point where the lines intersect.

Theorem 5: Composition of Rotation

The composition for two rotations with respect to the same point is again a rotation.

Theorem 6: Composition of Translation

The composition of two translations is also a translation.

Composition of translation examples

Let's look at a worked example regarding the composition of translations.

Composition, composition of rotation, StudySmarter Composition of rotation, StudySmarter Originals

Find the composition in the given rotation.

Solution: Here, the rotation degree for arc PP' is given as $90 °$ and the rotation degree for arc P'P'' is given as $45 ° .$ Then, for the rotation degree of arc PP'', we use the composition of rotation. This simply means that we add the degrees of rotation for both of the two rotations to obtain our composition of rotation.

$\Rightarrow (R C, 90 ° \circ R C, 45 °) = (R C, 135 °)$

Composition of functions

If two functions $f (x)$ and $g (x)$ are given, we can combine the functions to create a new, composite function. This means that we take function g and use it as an input for the other function f. Then, the composition of these functions is denoted as $f (g (x))$ or $(f \circ g) (x) .$ We call it "f of g of x."

We can also form a composite function using a reverse order, where we consider function f as the input for function g. In this case, we represent it the composite function as $g (f (x))$ or $(g \circ f) (x) .$

Note that $g (f (x))$ and $f (g (x))$ will yield different functions unless they are perfect inverse (opposite) of each other in which $f (g (x)) = g (f (x)) = x .$

Order of composition of functions

When dealing with composition, the order of the functions f and g is important to note and should be kept in mind. Here an example is presented to understand this concept better.

Consider two functions $f (x) = x^{2}$ and $g (x) = x + 2 .$ Now, we will find the compositions $f \circ g$ and $g \circ f$ and check whether they give the same answers or different ones.

$f \circ g (x) = f (g (x)) = f (x + 2) = {(x + 2)}^{2} = x^{2} + 2 x + 4$

In the above equation, we use function g as an input for function f. Now, let us consider a composite function in which we use function f as the input for function g.

$g \circ f (x) = g (f (x)) = g (x^{2}) = x^{2} + 2$

We can see the difference between both compositions. In other words, $f \circ g (x) d o e s n o t e q u a l g \circ f (x)$ .

Domain of composite functions

All the values which are accepted by the particular function are called the domain of that function. However, for a composition of functions, we are dealing with two different functions. So, it is necessary to consider both functions when dealing with the domain of a composite function.

There are times when we cannot compose some particular functions for all the values of x. So, in this situation, we will restrict some values of x in order to use composition. Those restricted values become the domain for that function. In compositions, the domain of the second function can either be the same as the first function or lie within it.

We consider two functions $f (x) = 4 x - 6, g (x) = x .$ Here let's find the domain for the composition $g (f (x)) .$

As function g is the square root, we can only consider greater than or equal to zero values for x.

Now for the function f, it will become the input of function g. So f should have values greater than or equal to zero.

So $f (x) \geq 0$ will be only possible when the value x as $x \geq \frac{3}{2}$ is considered.

Here the value of x is obtained as:

$f (x) \geq 0 \Rightarrow 4 x - 6 \geq 0 \Rightarrow 4 x \geq 6 \Rightarrow x \geq \frac{6}{4} \Rightarrow x \geq \frac{3}{2}$

Hence, the composition function $g (f (x))$ has a domain of $x \geq \frac{3}{2} .$

Range of composite functions

The range of a function concerns the output values of that function. We can calculate the range of the composite function just as we do for the range of any other function. The range of the composite function can either be the same as the second function (i.e. outer function) or lie within it.

Let $f (x) = x - 8, g (x) = x^{2} .$ We will find the composition $g (f (x)) .$ Here the function f can take any value. But the second function g is in the form of the square. So the values will be greater than equal to zero. Now for the composition $g (f (x))$ we have:

$g (f (x)) = g (x - 8) = {(x - 8)}^{2} = x^{2} - 16 x + 64$

As f can be anything and applying g to it, where g can be greater than or equal to zero, the range of the composition $g (f (x))$ will be greater than or equal to zero.

Composition of functions examples

Let's better understand the composition of functions with some worked examples.

Two functions $f (x) = 3 x$ and $g (x) = 2 x^{2} - 5$ are given. Find the composition $(g \circ f) (x) .$

Solution: Here we have to find $g \circ f .$ So we will first consider function and then we will use it as input for the function g. So we can write it as:

$(g \circ f) (x) = g (f (x)) = g (3 x) = 2 {(3 x)}^{2} + 5 = 2 (9 x^{2}) + 5 = 18 x^{2} + 5$

Find $f (g (- 1))$ for the functions $f (x) = x^{2} - 2 x$ and $g (x) = x - 5 .$

Solution: We are asked to calculate the composition of f and g when the value of x is given as $(- 1) .$ First, we use function g as an input for function f. To do this, we replace the variable x in function f with the entire function of g, that is id="2728522" role="math" $g (x) = x - 5$ .

Note: The input function goes into every x value and not just in the first value.

$f (x - 5) = {(x - 5)}^{2} - 2 (x - 5) = x^{2} - 10 x + 25 - 2 x + 10 = x^{2} - 12 x + 35$

For the last step, we simply input the value of x, -1, in the above composite equation.

$f (g (x)) = x^{2} - 12 x + 35 ∴ f (g (- 1)) = {(- 1)}^{2} - 12 (- 1) + 35 = 48$

Alternatively, we can also use the following approach to solve the composition. First, we evaluate $g (x)$ :

$\Rightarrow g (x) = x - 5 ∴ g (- 1) = - 1 - 5 = - 6$

Then, we proceed further by using the calculated value above as an input for the function f.

$f (g (- 1)) = f (- 6) = x^{2} - 2 x = {(- 6)}^{2} - 2 (- 6) = 36 + 12 = 48$

$∴ f (g (- 1)) = 48$

Composition - Key takeaways

A composition of functions is a mathematical process where two functions are combined to generate a composite function. The output of one function is used as the input of another.
A composition of transformations is a process where a combination of transformations (translations, rotations, reflections, etc.) is performed on a shape or figure consecutively, with the resulting transformed shape from one transformation used as the starting point in the following transformation.
When dealing with composition, the order of functions f and g is important.
In composition, the domain of the second function can either be the same as the first function or lie within it.
The range of the composite function can either be the same as the second function (i.e. outer function) or lie within it.

Frequently Asked Questions about Composition

Composition is the combination of two functions or transformations.

The rule of composition is that the output of one function should be input for another function.

We can write composition for functions f(x) and g(x) as f(g(x)). And we call it as "f of g of x."

We can find the composition by substituting one function into another, regarding it as an input.

We use composition of transformations to create a new image (or figure) using transformations, whereby a combination of transformations is performed on a shape or figure consecutively, using the resulting transformed shape from one transformation as the starting point in the following.

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Flashcards in Composition21 Start learning

What is an identity map?

An identity map is a function that gives back the value that is put in meaning that the input and the output are the same.

What are the other names used for identity map?

It is also referred to as identity function, identity relation, identity operator and identity transformation.

What does a map mean in identity map?

A map shows the relationship between two sets of elements.

What are some properties of an identity map/function?

The elements in the domain and co-domain of the map are the same.
The graph of an identity function is a straight line with a slope of 1.

Is the following function an identity function? Why or why not?

Input	-3	-2	-1	0	1	2	3
Output	-6	-4	-2	0	2	4	6

No, it is not an identity function because the input and output are not identical.

The identity map has a matrix called the ________matrix.

identity

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Meaning of composition

Composition of transformations