**Introduction to Composition of Functions**

Suppose that *f* and *g* are functions and that the domain of *g* contains the range of *f* . This means that if *x* is in the domain of *f* then *f* ( *x* ) makes sense but also *g* may be applied to *f* (*x*) (Fig. 1.46). The result of these two operations, one following the other, is called *g composed* with *f* or the *composition* of *g* with *f* . We write

**Examples**

**Example 1**

Let *f* ( *x* ) = *x* ^{2} − 1 and *g* ( *x* ) = 3 *x* + 4. Calculate *g ο f* .

**Solution 1 **

We have

Notice that we have started to work *inside the parentheses:* the first step was to substitute the definition of *f* , namely *x* ^{2} − 1, into our equation.

Now the definition of *g* says that we take *g* of *any* argument by multiplying that argument by 3 and then adding 4. In the present case we are applying *g* to *x*^{2} − 1. Therefore the right side of equation (*) equals

3 · ( *x*^{2} − 1) + 4.

This easily simplifies to 3*x*^{2} + 1. In conclusion,

*g ο f* ( *x* ) = 3*x*^{2} + 1.

**Example 2**

Let *f* ( *t* ) = ( *t* ^{2} − 2)/( *t* + 1) and *g* ( *t* ) = 2*t* + 1. Calculate *g ο f* and *f ο g* .

**Solution 2 **

We calculate that

We compute *g* of any argument by doubling it and adding 1. Thus equation (**) equals

One of the main points of this example is to see that *f ο g* is different from *g ο f* . We compute *f ο g* :

So *f ο g* and *g ο f* are different functions.

**You Try It:** Let *f* ( *x* ) = ∣ *x* ∣ and . Calculate *f ο g* ( *x* ) and *g ο f* ( *x* ).

We say a few words about *recognizing* compositions of functions.

**Examples**

**Example 3 **

How can we write the function *k* ( *x* ) = (2 *x* + 3)^{2} as the composition of two functions *g* and *f* ?

**Solution 3 **

Notice that the function *k* can be thought of as two operations applied in sequence. First we double and add three, then we square. Thus define *f* ( *x* ) = 2*x* + 3 and *g* ( *x* ) = *x*^{2} . Then *k* ( *x* ) = ( *g ο f* )( *x* ).

We can also compose three (or more) functions: Define

( *h ο g ο f* )( *x* ) = *h* ( *g* ( *f* ( *x* ))).

**Example 4**

Write the function *k* from the last example as the composition of three functions (instead of just two).

**Solution 4 **

First we double, then we add 3, then we square. So let *f*(*x*) = 2*x* , *g*(*x*) = *x* + 3, *h*(*x*) = *x*^{2}. Then *k**(x*) = ( *h ο g ο f* )(*x*).

**Example 5**

Write the function

as the composition of two functions.

**Solution 5 **

First we square *t* and add 3, then we divide 2 by the quantity just obtained. As a result, we define *f* ( *t* ) = *t* ^{2} + 3 and *g* (*t*) = 2/ *t*. It follows that *r* (*t*) = ( *g ο f* )(*t*).

**You Try It:** Express the function *g* ( *x* ) = 3/(*x*^{2} + 5) as the composition of two functions. Can you express it as the composition of three functions?

Practice problems for this concept can be found at: Calculus Basics Practice Test.

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