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:
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:
This new function will be a composite function and is represented by writing a small circle between the two functions as follows:
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:
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:
It reads g composed with f and is equivalent to writing that the function f depends on the function g:
Let’s see it with an example to make it clearer:
With the two previous functions:
Let’s get the f function composed with g:
And the function g composed with f:
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):
Now, in the place where it is f(x), we replace it with the expression of the function:
And finally, in function g(x), we replace x with the expression of function f(x) and we are left:
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:
We replace g(x) with its expression:
And finally, in function f(x), we substitute x with the expression of g(x):
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:
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:
f composed with g:
f composed with g is equal to function g, which depends on function f(x):
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)
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:
g compound with f:
g composed with f is the function f that depends on the function g(x):
Therefore, we substitute g(x) for its expression in the place where x in f(x) corresponded to it:
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):
We replace f(x) with its expression in the place where the x appears:
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:
We group terms and finally, the denominator x+1 that we have in numerator and denominator is cancelled:
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:
We replace x with the expression of g(x) in the same function g(x):
And finally we simplify:
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.