Composite Functions Made Clear and Examples

Composite functions is the fancy name given to functions whose argument is also a function. Let's say I give you the functions defined by the rules:

Definitions of two functions

Here we can create a new function, using g(x) as the argument:

A composite function

In this case we substitute the "x" in sin(x) by x squared to get:

A composite function

This new composite function is written as:

Function composition notation

And the rule is:

Expression of the composite function

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Now, a question for you:

What is the domain of f circle g?

Remember that we need to specify the domain to have a function properly defined.

In this case, f circle g is defined for all valid values of g(x). Why? Because g(x) is the argument!

But what are the valid values of g(x)? sin(x) is well defined for all real values of x. So, the domain is the set of real numbers.

Let's do some examples...


Example 1

Let's say we have the functions:

Three functions

We want to find:

Example 1 of composite function

We want the value of S circle P at y.

According to our definition of composite functions:

Example 1 of composite function

Now it looks simpler, doesn't it? What would our P(y) be? Well, just replace x by y:

Replacing variable x by variable y

So, we have:

Finding the expression of the composite function

Now, we just replace the x in:

Definition of squaring function

by 2 to the y. So, we get:

Replacing into the squaring function

And that is:

Final expression of example 1 of composite function

And that's our answer.


Composition of More Than Two Functions

Let's keep working the functions:

Three functions

Now, let's say we want to find:

Composition of three functions

Because we have the (t) at the end, what we wanto to find is the value of the function at t. According to our definition:

Finding the expression of the composition of three functions

In the previous examples, we've found that:

What we've got in the previous example

Here, instead of why we have s(t). So, we just to replace and get:

Applying the previous result with our new variable

But what is s(t)? It is:

Changing letters so everything matches up

So, finally:

The final expression of the composite function

What would had happened if we instead were asked to find:

What do we get if we change the placement of the parenthesis?

What do you think? Will it be the same result? Let's find out:

Finding the expression of the composite function

We first need to find:

Going by parts

We again apply our definition:

Finding the expression of composite function

And we know that:

Definition of our function

So, replacing in the previous equation:

Finding the expression of the composite function

But, we also know:

The expression of the outer function

So, we have:

Finding the expression of the composite function

And replacing that in what we were originally looking for:

Finding the expression of the composite function

And the function S just squares whatever it has as argument:

Final expression of the composition of three functions

And it is the same result. This is a property of all composite functions:

No matter where you put the parenthesis, you get the same value.

That is, using function notation:

No matter where you put the parenthesis, you get the same value.

Because these two are equal, we usually just write:

The composition of three functions


Example 3

Now we'll solve the reverse of what we were doing previously. Let's say we have the function defined by:

Another function

And we want to express f in terms of our functions S, P and s:

Our three initial functions

We can solve this in two ways, but getting the same result. We have:

Our new function

We can start analyzing the "outer" function sin. We have a function sin(something). Our s(x) is defined as sin(x), so we have:

The outer function is sin

Ok, now we have to express 2 to the x using our functions. This one is easy. We already have:

Expression of our new function

So,

Using the definition of P(x)

Writing this in function notation:

Our function in composition notation

What is the othe way of doing this problem? You could also had started repalcing first the 2 to the x. As an exercise, you might want to do the same with the function:

Additional exercise


Conclusion

  • A composite function is a function whose argument is another function.
  • When you have the composite of three functions, no matter where you put the parenthesis, you get the same function.
  • Composite functions problems are not hard. You just need to get used to notation.

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