the composition of functions mathematics

Introduction

The composition of functions refers to the process of the combination into a single function of two or more functions. The essential idea is that the function’s input is not numerical. Instead, the input here is another function. In mathematics, a function performs a set of operations on an input to produce an output. Therefore, the composition of functions takes place when one function’s output becomes another function’s input. The study material notes on the composition of functions will make you familiar with this topic efficiently.

Order of composition

The Order in which the composition of functions takes place makes a big difference to the result. That is why we must focus on these study material notes on the composition of functions. 

For functions whose representation takes place by f(x) or g(x), the representation of composition shall take place by f(g(x)) or g(f(x)). You should know that f(g(x)) does not usually have a value that is similar to the value of g(f(x)). This is why the Order is so important when undergoing the calculation of its composition. Always start with the innermost function and follow this habit as a rule.

Now, let us consider f(x) = x2 and g(x) = x + 3. In the composition of functions study material example, f(g(x)) is ascertained by taking g(x) and applying f to all of it. The g(x) here is x + 3. This provides us with the following:

 fg(x) = f(x + 3) = (x + 3)2 = x2 + 6x + 9. 

This is not the same as gf(x), due to the fact that gf(x) = x2 + 3 

This, generally speaking, does not equal x2 + 6x + 9.

Decomposing Functions

Sometimes, the writing of a particular function can take place as the composition of two other functions. This is what mathematicians call decomposing functions.

So let’s study the following function of the composition of functions study material,

f:R → R by f(x) = (3x + 2)3

Now, the denotation of the inner function takes place by h. 

As such, h:R → R by h(x)=3x+2

Moreover, g:R → R by g(x) = x3. 

(g∘h)(x) = g(h(x)) 

Carrying on solving, = g(3x+2)

So, we get = (3x+2)3

= f(x) 

Furthermore, you will notice that g∘h = f. Therefore, the decomposition of the function f takes place. The writing of the function f can take place in other ways as a composition of two functions. However, the way that has been explained above goes properly with the chain rule. As such, the chain rule gives

f′(x) = (g∘h)′(x)

Continuing solving = g′(h(x))h′(x)

So, we get = 3(h(x))2⋅3

Finally, we get = g(3x+2)2

Understanding of the Composition Of Functions

Function composition is the only way to establish existing functions. The composition of functions takes place by doing certain operations with the function outputs. This way, the result gets defined as the new function’s output. 

Suppose an addition has to be made of two columns of numbers. These columns are representative of the separate annual incomes of a husband and wife for a certain number of years. The result here would be the total household income of this couple. 

Suppose this has to be performed every year. This can be done by adding a particular year’s income. Afterward, the data can be collected in a new column. 

Let w(y) be the wife’s income, and h(y) be the income of the husband in year y. Now, let the total income be represented by T.

So, the definition of the new function can take place as:

T(y) = h(y) + w(y)

If this is the case for every year, then one can emphasize the relationship between the functions without reference to the year. Now, the writing can take place as

T = h + w

In case you are dealing with two functions, f(x) and g(x), that are characterized with actual number outputs, the defining of the following new functions has to take place:

• F + g
• F − g
• f⋅g
• f + g

This defining can take place by the relations as follows:

• (f + g)(x) = f(x) + g(x)
• (f − g)(x) = f(x) − g(x) 
• (f⋅g)(x) = f(x)⋅g(x) 
• (f/g)(x) = f(x)/g(x)

Conclusion

The composition of functions refers to the combination of two or more functions. Here, the function’s input is not a numerical value but another function. So, one function’s output becomes another function’s input. It is essential to understand the Order of composition and its decomposing functions. One also needs to study understanding the composition of functions.