Section 1.4- Composition Combinations 

COMPOSITION COMBINATIONS

The composition combination of functions is similar to that of arithmetic combinations, most commonly used for operations such as sale tax. Involving at least two functions, composition combinations involve the input of one function into another function. Unlike arithmetic combinations, which involve using arithmetic operations in order to combine functions, composition combinations use a function as an input for another function, replacing variables.

The main type of composition combination is as follows:

Composition Function

 Here is an example of the composition function:

Example Function

Now combine f(x) and g(x) by replacing the variable in f(x) with g(x)

Now simplify and find the answer

If you were to graph this function, it would look like this

 Domain and Range of Composite Functions

Composite combinations involve the combining of two functions, which means that when graphing the end simplified equation the other functions must also be possible. This means that the range and domain of all equations can only be graphed and technically the only correct answers. This occurs when impossible solutions are possible in later equations. This is explained in the following picture.

All impossible answers are also in impossible in the end equation no matter what transformation occurs. Here's an example:

Composition Domain and Range

 The domain of f(x) is all real numbers, while the domain of g(x) is greater than or equal to 1 or -1. Now lets finish the equation.

The end composite equation's domain is all real numbers, yet because the g(x) equation has a domain greater than or equal to 1 or -1, the entire composition equation must have a domain greater than or equal to one.