Composition of functions. Composite function properties. Solved exercises.

Next I’m going to explain what the composition of functions consists of in an easy way, with exercises solved step by step. I will explain what a composite function is and what its properties are.

What is a composite function

Normally, a function depends on a single variable, such as for example:

composicion de funciones ejercicios resueltos

A composite function is a function that instead of depending on x, depends on another function.

If we have two functions f and g, a new function can be defined, such that function g will depend on function f:

Composition of functions exercises

This new function will be a composite function and is represented by writing a small circle between the two functions as follows:

function composition exercises

This expression is read f composed with g. We have to be very careful because we read the reverse of how the functions are composed, because the function g is really composed by the function f.

The function behind is the one inside the other function:

compound functions solved exercises

Of course, the function composed between f and g can also be formed, so that f depends on function g. It would be written as follows:

composite function exercises

It reads g composed with f and is equivalent to writing that the function f depends on the function g:

compound functions exercises

Let’s see it with an example to make it clearer:

With the two previous functions:

Solved function composition exercises

Let’s get the f function composed with g:

composicion de funciones ejercicios resueltos

And the function g composed with f:

functions f or g exercises solved

We start with compound f with g.

Compound f with g is equal to function g which depends on function f (remember that it reads backwards):

compound function exercises

Now, in the place where it is f(x), we replace it with the expression of the function:

exercises function composition

And finally, in function g(x), we replace x with the expression of function f(x) and we are left:

function composition exercises

We are going to repeat it again, but in this case to obtain the compound function g composed with f.

g compound with f, is the function f that depends on the function g:

exercises of resolved composite functions

We replace g(x) with its expression:

funcion compuesto ejercicios

And finally, in function f(x), we substitute x with the expression of g(x):

composicion de funciones ejercicios
As you have seen, the composition of functions does not have the commutative property, since g composed with f is not equal to f composed with g:

composicion de funciones ejemplos

funcion compuesto ejercicios resueltos

Resolved exercise of composition of functions

We are going to solve a more complete function composition exercise step by step, so that this concept of function composition becomes much clearer to you.

Let the functions be:composicion de funciones

Calculate:

combination of functions exercises solved

f composed with g:

f composed with g is equal to function g, which depends on function f(x):

f or g example functions

We replace f(x) with its expression and then in function g(x), instead of placing x, we write the expression of function f(x)

fog y gof ejercicios resueltos

Up to this point we would have obtained the composite function. Now let’s operate to remove parentheses and simplify the expression.

We multiply the number by the parentheses and get a common denominator to add both terms:

exercises of composition of solved functions

g compound with f:

g composed with f is the function f that depends on the function g(x):

Solved Function Composition Exercises

Therefore, we substitute g(x) for its expression in the place where x in f(x) corresponded to it:

exercises composite functions

And in this case, simplifying is much simpler, since we only have to solve the parentheses and regroup terms:

f composite with f:

Is it possible to obtain a composite function with the same function?

Well, you’ll see. f composed of f is the function f, which depends on the same function f(x):

funcion compuesto ejercicios resueltos paso a paso

We replace f(x) with its expression in the place where the x appears:

exercises solved function composition

We already have the composite function. Now we are going to simplify the expression we have left. To do this, we first solve the parentheses and get a common denominator in both the numerator and the denominator:

examples of function composition

We group terms and finally, the denominator x+1 that we have in numerator and denominator is cancelled:

solved exercises of compound function

g composed with g:

We are going to obtain the compound function g composed with g. which is the function g that depends on the same function g:

exercises of resolved composite functions

We replace x with the expression of g(x) in the same function g(x):
composite function exercises

And finally we simplify:

function composite examples

You will have noticed that the process is always the same. You must be very attentive to what it is or what you have to replace in each case and once you get the composite function, the difficulty is to simplify the expression.

You only have to be careful in composing the function upside down, since the expression is read the opposite of how it is formed.