, in Figure 1:

gf(x) = ((x-6/3) - 1 = x - 6 - 6 = x - 12

When x = 0, gf(x) = -2

When x = -3, gf(x) = -2.5

References

Department of Mathematics, Sinclair Community College, Dayton, OH.. Composite Functions and their Domains. Retrieved 18 Feb 2014 from http://www.sinclair.edu/centers/mathlab/pub/findyourcourse/worksheets/Algebra/CompositeFunctionsAndTheirDomains.pdf

Henri Picciotto. Function Diagrams. Retrieved 18 Feb 2014 from http://www.mathedpage.org/func-diag/

Pierce, Rod. (26 Oct 2012). "Composition of Functions". Math Is Fun. Retrieved 19 Feb 2014 from http://www.mathsisfun.com/sets/functions-composition.html

Stapel, Elizabeth. "Functions: Domain and Range." Purplemath. Retrieved 19 Feb 2014 from http://www.purplemath.com/modules/fcns2.htm.

Stapel, Elizabeth. "Inverse Functions and Composition." Purplemath. Retrieved 19 Feb 2014 from http://www.purplemath.com/modules/fcns6.htm.

Wee Wen Shih. (21 Dec 2011). "How to find range of a composite function" [Slideshare]. Retrieved 19 Feb 2014 from http://www.slideshare.net/weews/how-to-find-the-range-of-a-composite-function

Composite Functions

Applying one function to the results of another.

When the result of f(x) is sent through g(x), it can be represented as (g º f)(x) or gf(x).

_____ _______ _____

2 6 6

Figure 1

Domain and Range

Domain

: Complete set of possible values of the independent variable as long as the function is valid, i.e. One-to-one or many-to-one function.

For example,

For f(x) = x , x can be all Real values to form a valid function.

However, for f(x) = √x , the value of

x

should a positive number in order for f(x) to be a real value. Therefore, x>0 is the domain.

Range

: Complete set of all possible values of the dependent variable.

For example,

Using back f(x) = x . For all real values of x, f(x) will be a positive value. Therefore the range: f(x)>0 .

For f(x) = √x . For x>0, positive values of f(x) can only be obtained. When f(x)<0, f(x) = √x will not be a valild function (one-to-many). Hence, f(x)>0 is the range.

Step 1

Find the domain of g(x).

Function g(x) cannot pick up the value 3. Consequently, the composition also cannot pick up the value 3.

Step 2

Using the value(s) of g(x) for which function f(x) cannot pick up, find the domain of g(x) which results in a value of g(x) that function f(x) cannot pick up.

The answers coming out of function g(x) come out in the form x/(x-3). Since function f(x) cannot pick up -2, we must lookout for any values of x that cause x/(x-3) = -2. This is because these values create an answer that cannot progress through the composition.

x = -2 => x=2

The domain of fg(x) is all real values with the exception of x=3 and x=2.

2

In order to understand and find out the domain and range of composite functions, these fundamental ideas must be understood.

Domain and Range of Composite Functions

**Domain**

f(x) = 1 and g(x) = x

What is the domain of fg(x)?

____ _____

x + 2 x - 3

Linear Composite Functions

Example:

f(x) = x + 4

g(x) = x + 1

gf(x) = g(x + 4) = (x + 4) + 1 = x + 5

Quadratic Composite Functions

Inverse Functions and Composite Function

___

x-3

Step 1

Find the domain of the "inside" (input) function. If there are any restrictions on the domain, keep them.

Step 2

Construct the composite function. Find the domain of this new function. If there are restrictions on this domain, add them to the restrictions from Step 1. If there is an overlap, use the more restrictive domain (or the intersection of the domains). The composite may also result in a domain unrelated to the domains of the original functions.

**Range**

**Example 1:**

Step 1:

Find the range of the input function.

Step 2:

Put this range on the x-axis of the graph of the second function.

Step 3:

Find the corresponding range.

Step 1

Domain of g(x) is x ≤ 3.

Step 2

The domain for fg(x) = 5 - x is all real numbers. Since there is an overlap, the input function's domain which is more restricted is the domain of the composite function.

The domain of fg(x) is x ≤ 3.

If f(x) = x + 2 and g(x) = √(3 - x) , then f(g(x)) = (√(3 - x)) + 2 = 5 – x.

What is the domain of fg(x)?

**Example 2:**

2

2

**Example**

Suppose g(x) = x + x + 1 , x is a member of all real values and h(x) = x + (x/4) , x > 0

2

Find the range of hg.

Step 1:

Minimum point of g(x) is ( -0.5 , 0.75 ).

Therefore, g(x) ≥ 0.75

Step 2&3:

Range of g(x)

Range of hg(x)

y

x

**Therefore, hg(x) ≥ 4**

2

2

2

2

Given f (x) = 2x – 1 and g(x) = (1/2)x + 4,

Find f (x), g (x), [(fg) (x)], and (g f )(x).

-1

-1

-1

-1

-1

For f (x):

-1

For g (x):

-1

For g f (x):

-1

-1

For [(fg) (x)]:

-1

Conclusion:

**Composite Functions, Domain and Range**

**By Sher Lynn, Tarendran and Jing Jun**