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:

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

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

This new composite function is written as:

And the rule is:

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:

We want to find:

We want the value of S circle P at y.

According to our definition of composite functions:

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

So, we have:

Now, we just replace the x in:

by 2 to the y. So, we get:

And that is:

And that's our answer.

Composition of More Than Two Functions

Let's keep working the functions:

Now, let's say we want to find:

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:

In the previous examples, we've found that:

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

But what is s(t)? It is:

So, finally:

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

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

We first need to find:

We again apply our definition:

And we know that:

So, replacing in the previous equation:

But, we also know:

So, we have:

And replacing that in what we were originally looking for:

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

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:

Because these two are equal, we usually just write:

Example 3

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

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

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

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:

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

So,

Writing this in function 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:

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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