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### Hawkes Learning Systems:College Algebra

Section 4.5: Combining Functions

Objectives

- Combining functions arithmetically.
- Composing functions.
- Decomposing functions.

Combining Functions Arithmetically

Addition, Subtraction, Multiplication and Division of Functions

1.

2.

3.

4.

The domain of each of these new functions consists of the common elements (or the intersection of elements) of the domains of f and g individually.

Example: Combining Functions Arithmetically

Given that solve:

a.

Remember that

Continued on the next slide…

Example: Combining Functions Arithmetically

Given that find a. and b.

a.

b.

Remember that

Remember that

Composing Functions

Composing Functions

Let f and g be two functions. The composition of f and g, denoted , is the function defined by .

The domain of consists of all x in the domain of g for which g(x) is in turn in the domain of f. The function is read “f composed with g,” or “f of g.”

Composing Functions

Caution!

Note that the order of f and g is important. In general, we can expect the function to be different from the function . In formal terms, the composition of two functions, unlike the sum and product of two functions, is not commutative.

Composing Functions

The diagram below is a schematic of the composition of two functions. The ovals represent sets, with the leftmost oval being the domain of the function g. The arrows indicate the element that x is associated with by the various functions.

Example: Composing Functions

Given f(x) = x2and g(x) = x + 5 , find:

a.

= 112

= 121

First, we will find g(6) by replacing x with 6 in g(x).

Next, we know that f composed with g can also be written . Since we already evaluated g(6), we can insert the answer to get f(11).

Continued on the next slide…

Example: Composing Functions (cont.)

Given f(x) = x2+ 2 and g(x) = x + 5 , find:

b.

= (x + 5)2 + 2

= x2 +10x + 25 + 2

= x2 +10x + 27

Again, we know by definition that .

Note: since we solved for the variable x we should be able to plug 6 into x and get the same answer as in part a. Verify this.

Example: Composing Functions

Let f(x) = x – 6 and g(x) = . Simplify the composition and find the domain for:

= g(x – 6)

=

Domain: [6, )

The domain of must be any x such that x – 6 > 0 since x - 6 is under a radical.

Decomposing Functions

Often functions can be best understood by recognizing them as a composition of two or more simpler functions. For example, the function can be thought of as the composition of two or more functions.

Note: if then:

Decomposing Functions

Ex: The function can be written as a composition of functions in many different ways. Some of the decompositions of f(x) are shownbelow:

a.

b.

c.

Example: Decomposing Functions

Decompose the function into:

a. a composition of two functions

b. a composition of three functions

Note: These are NOTthe only possible solutions for the decompositions of f(x)!

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