In this Lesson, we will be looking at Composite function.

Let’s say we have a function f (x) = 2x + 3,

we can be asked to find f (2).

All we will do is to replace the x by the 2.

So we have

f(2) = 2 (2) + 3

f(2) = 7

 

Solving f(g(x)) - read as F of G of x

Let’s say we have another function

g(x) =x^2 + 1. We can be asked to find f (g(x)).

The same way we put the 2 in place of the x, we must put the g(x) in place of the x.

This becomes

f (g(x)) = 2  (g(x)) + 3.

Now we know what g(x) is, so we can put the x^2 + 1 to replace the g(x).

So we have

f (g(x)) = 2 (x^2 + 1) + 3. This is what composite function is all about.

Let’s expand this.

f (g(x)) = 2 x^2 + 2 + 3

f (g(x)) = 2x^2 + 5

 

Solving g(f(x)) - read as G of F of x

Now let’s do g(f(x)). We can write it this way to make it clearer. This time we are putting the f(x) in g(x).

g(f(x)) = (f(x))^2 + 1

We know that f(x) = 2x + 3. we put that in place of the f(x) to get 

g(f(x)) = (2x + 3)^2 + 1

g(f(x)) = (2x + 3) (2x + 3) + 1

g(f(x)) = 4x^2 + 12x + 10

There is really no limit to how the question can be, but the idea is the same.

 

Solving f(f(x)) - read as F of F of x

Let’s do f(f(x)). We are putting f(x) back into f(x).

so we will have

f(f(x)) = 2 (2x + 3) + 3.

We can simplify to get

f(f(x)) = 4x + 9.

 

Solving f(g(3)) - read as F of G of 3

Let’s look at f(g(3). For a question like this, we will solve for the f(g(x) and then replace our x with 3.

We’ve already done f(g(x),

f (g(x)) = 2x^2 + 5

Let’s now put the 3 in place of the x.

f (g(3)) = 2(3)^2 + 5.

f (g(3)) = 23.

Let’s end this Lesson here. In the next Lesson, we will be looking at operations on functions. Have a great day. Good bye