Composite Functions

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Composite functions are operations that take two or more functions as one function such as. This is mainly to do with taking numbers from one set to another set.

So, for example, if a function took a number from set A to B and another function took a number from set B to C, the composite would take a number from set A directly to C.

Here is a diagram showing how the functions and can transform an input to an output.

Composite functions, An image which illustrates composite functions, StudySmarter An image which illustrates composite functions

Values in sets can be mapped to other sets by composite functions.

Properties of composite functions

There are important properties of composite functions that we should remember:

Property name	Definition
Associative	If functions are composable, they are always associative. This means that no matter where the brackets are located in a function, it makes no difference to the overall function result. Therefore if are composable then.
Commutative	If functions are composable it means they are not necessarily commutative. Commutativity is when swapping the order of the function's composition, does not affect it, e.g. (ab=ba).
One-to-one	A one-to-one composite function is where there is a single output for every input. There is also a many-to-one function where many inputs can render the same output. In the definition of a function, no composite function can be one-to-many.
Inverse	An inverse must exist for a composite function, therefore, there cannot be an output for which there does not exist an input.

How do we find a composite function?

Essentially, we are carrying out a function of a function. Let's say we're trying to find.

We would firstly take (the output of ) and then use it as an input in hence getting. Let's look at a worked example.

Examples of composite functions

and . If , Find the value of.

Steps	Example
Step 1: Rewrite h (x).	.
Step 2: Find the output of the inside function first.
Step 3: Substitute this newly found output as the input in the outside function.
FINAL ANSWER

and

. If

, find

Steps	Example
Step 1: Rewrite h (x).
Step 2: Find the output of the inside function first.
Step 3: Substitute this newly found output as the input in the outside function.	`
FINAL ANSWER

. Find .

Steps	Example
Step 1: rewrite .
Step 2: Find the output of the inside function first.
Step 3: Substitute this newly found output as the input in the outside function.
FINAL ANSWER

What are some harder composite functions?

Sometimes quadratic, trigonometric and reciprocal functions can come into play, however, the logic is exactly the same as with the easier linear examples we have seen above. Let's look at some more worked examples.

Find the value of .

steps	Example
Step 1: Rewrite h (x).
Step 2: Find the output of the inside function first.
Step 3: Substitute this newly found output as the input in the outside function.
FINAL ANSWER

, with Find the value of .

steps	Example
Step 1: Rewrite h (x).
Step 2: Find the output of the inside function first.
Step 3: Substitute this newly found output as the input in the outside function.
FINAL ANSWER

. . Find

steps	Example
Step 1: Rewrite h (x).
Step 2: Find the output of the inside function first.
Step 3: Substitute this newly found output as the input in the outside function.
FINAL ANSWER

Composite Functions - Key takeaways

Composite functions are when a combination of functions are undertaken to give a common output.
Always perform the inside function first before inputting this value into the outside function.

Frequently Asked Questions about Composite Functions

A composite function is a composition of two or more functions.

By looking at the domain of the first function applied.

By applying a singular input and finding the output of the functions put together.

By getting an output from inputing a value into the original function, then applying all the other functions to this output value.

Final Composite Functions Quiz

Question

f(x) = x - 1 , g(x) = x + 1, Find fg(2)

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Answer

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Question

f(x) = 3x + 5, g(x) = 5x - 2. Find fg(6)

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Answer

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Question

f(x) = 7x + 7, g(x) = x/7. Find gf(4).

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Answer

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Question

g(x) = x - 3 , f(x) = 4x + 5. Find fg(8).

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Question

f(x) = 7x - 2, g(x) = sin(x). Find fg(x).

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Answer

7cos(x) - 2.

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More about Composite Functions

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Composite Functions

Composite Functions

Properties of composite functions

How do we find a composite function?

Examples of composite functions

What are some harder composite functions?

Composite Functions - Key takeaways

Frequently Asked Questions about Composite Functions

What is a composite function?

How do you find the domain of a composite function?

How do we compose two functions?

How do we find the value of a composite function?

Final Composite Functions Quiz

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