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Function Composition & Inverse Functions: Understanding Notation & Role Swapping, Lecture notes of Algebra

The concept of function composition and inverse functions using the example of a quadratic function. It covers the basics of function notation, the process of composing functions, and the significance of inverse functions. The document also includes examples and visualizations to help clarify the concepts.

What you will learn

  • What is the difference between function composition and function multiplication?
  • How do you perform function composition?
  • How do you find the inverse function of a given function?

Typology: Lecture notes

2021/2022

Uploaded on 02/03/2022

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Imperial Valley College
Math Lab
Functions: Composition and Inverse Functions
FUNCTION COMPOSITION
In order to perform a composition of functions, it is essential to be familiar with function notation.
If you see something of the form โ€œ๐‘“(๐‘ฅ)= [expression in terms of x]โ€, this means that whatever you see
in the parentheses following f should be substituted for x in the expression. This can include
numbers, variables, other expressions, and even other functions.
EXAMPLE: ๐‘“(๐‘ฅ)= 4๐‘ฅ2โˆ’13๐‘ฅ
๐‘“(2)= 4 โˆ™ 22โˆ’13(2)
๐‘“(โˆ’9)= 4(โˆ’9)2โˆ’13(โˆ’9)
๐‘“(๐‘Ž)= 4๐‘Ž2โˆ’13๐‘Ž
๐‘“(๐‘3)= 4(๐‘3)2โˆ’13๐‘3
๐‘“(โ„Ž + 5)= 4(โ„Ž + 5)2โˆ’13(โ„Ž + 5)
Etc.
A composition of functions occurs when one function is โ€œplugged intoโ€ another function.
The notation (๐‘“ โ—‹๐‘”)(๐‘ฅ) is pronounced โ€œ๐‘“ of ๐‘” of ๐‘ฅโ€, and it literally means ๐‘“(๐‘”(๐‘ฅ)).
In other words, you โ€œplugโ€ the ๐‘”(๐‘ฅ) function into the ๐‘“(๐‘ฅ) function.
Similarly, (๐‘” โ—‹๐‘“)(๐‘ฅ) is pronounced โ€œ๐‘” of ๐‘“ of ๐‘ฅโ€, and it literally means ๐‘”(๐‘“(๐‘ฅ)).
In other words, you โ€œplugโ€ the ๐‘“(๐‘ฅ) function into the ๐‘”(๐‘ฅ) function.
WARNING: Be careful not to confuse (๐‘“ โ—‹๐‘”)(๐‘ฅ) with (๐‘“ โˆ™ ๐‘”)(๐‘ฅ), which means ๐‘“(๐‘ฅ) โˆ™ ๐‘”(๐‘ฅ) .
EXAMPLES: ๐‘“(๐‘ฅ)= 4๐‘ฅ2โˆ’13๐‘ฅ ๐‘”(๐‘ฅ)= 2๐‘ฅ + 1
a. (๐‘“ โ—‹๐‘”)(๐‘ฅ)= ๐‘“(๐‘”(๐‘ฅ)) = 4[๐‘”(๐‘ฅ)]2โˆ’13 โˆ™ ๐‘”(๐‘ฅ)= 4(2๐‘ฅ + 1)2โˆ’13(2๐‘ฅ + 1)
= [๐‘ ๐‘–๐‘š๐‘๐‘™๐‘–๐‘“๐‘ฆ]โ€ฆ = 16๐‘ฅ2โˆ’10๐‘ฅ โˆ’ 9
b. (๐‘” โ—‹๐‘“)(๐‘ฅ)= ๐‘”(๐‘“(๐‘ฅ)) = 2 โˆ™ ๐‘“(๐‘ฅ)+ 1 = 2(4๐‘ฅ2โˆ’13๐‘ฅ)+ 1 = 8๐‘ฅ2โˆ’26๐‘ฅ + 1
A function can even be โ€œcomposedโ€ with itself:
c. (๐‘” โ—‹๐‘”)(๐‘ฅ) = ๐‘”(๐‘”(๐‘ฅ)) = 2 โˆ™ ๐‘”(๐‘ฅ)+ 1 = 2(2๐‘ฅ + 1)+ 1 = 4๐‘ฅ + 3
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Imperial Valley College

Math Lab

Functions: Composition and Inverse Functions

FUNCTION COMPOSITION

In order to perform a composition of functions, it is essential to be familiar with function notation.

If you see something of the form โ€œ๐‘“

= [expression in terms of x ]โ€, this means that whatever you see

in the parentheses following f should be substituted for x in the expression. This can include

numbers, variables, other expressions, and even other functions.

EXAMPLE : ๐‘“

2

2

2

2

3

3

2

3

2

Etc.

A composition of functions occurs when one function is โ€œplugged intoโ€ another function.

The notation (๐‘“ โ—‹๐‘”)(๐‘ฅ) is pronounced โ€œ๐‘“ of ๐‘” of ๐‘ฅโ€, and it literally means ๐‘“

In other words, you โ€œplugโ€ the ๐‘”(๐‘ฅ) function into the ๐‘“(๐‘ฅ) function.

Similarly, (๐‘” โ—‹๐‘“)(๐‘ฅ) is pronounced โ€œ๐‘” of ๐‘“ of ๐‘ฅโ€, and it literally means ๐‘”(๐‘“(๐‘ฅ)).

In other words, you โ€œplugโ€ the ๐‘“(๐‘ฅ) function into the ๐‘”(๐‘ฅ) function.

WARNING : Be careful not to confuse (๐‘“ โ—‹๐‘”)(๐‘ฅ) with (๐‘“ โˆ™ ๐‘”)(๐‘ฅ), which means ๐‘“(๐‘ฅ) โˆ™ ๐‘”(๐‘ฅ).

EXAMPLES : ๐‘“(๐‘ฅ) = 4 ๐‘ฅ

2

a. (๐‘“ โ—‹๐‘”)(๐‘ฅ) = ๐‘“(๐‘”(๐‘ฅ)) = 4 [๐‘”(๐‘ฅ)]

2

2

[

]

2

b. (๐‘” โ—‹๐‘“)(๐‘ฅ) = ๐‘”(๐‘“(๐‘ฅ)) = 2 โˆ™ ๐‘“(๐‘ฅ) + 1 = 2 ( 4 ๐‘ฅ

2

2

A function can even be โ€œcomposedโ€ with itself:

c. (๐‘” โ—‹๐‘”)

INVERSE FUNCTIONS

The notation for inverse functions can cause confusion. It is important to know that ๐’‡

โˆ’๐Ÿ

๐Ÿ

๐’‡(๐’™)

Instead, ๐‘“

โˆ’ 1

(๐‘ฅ) indicates the inverse function of ๐’‡(๐’™), which can be thought of as the function that

โ€œreversesโ€ ๐‘“(๐‘ฅ) , or โ€œundoesโ€ everything that ๐‘“(๐‘ฅ) does.

EXAMPLE : Let ๐‘“(๐‘ฅ) = 3 ๐‘ฅ โˆ’ 1.

In words , ๐‘“(๐‘ฅ) takes a number, multiplies it by 3 , and then subtracts 1 from it.

The opposite or reverse of this procedure would be to add 1 to a number, then divide it by 3.

For instance, ๐‘“(๐Ÿ’) = 3 โˆ™ 4 โˆ’ 1 = ๐Ÿ๐Ÿ. Input = 4 , Output = 11

In reverse , take the output 11, add 1 to it, then divide by 3: 11 + 1 = 12

12 รท 3 = ๐Ÿ’ = input.

In other words, the output became the input, and vice-versa. They switched roles!

Mathematically , In each ordered pair (๐‘ฅ, ๐‘ฆ) associated with a function, the x and y

โ€œswitch placesโ€: (๐‘ฅ, ๐‘ฆ) โ†’ (๐‘ฆ, ๐‘ฅ)

In the above example,

became

Procedure : EXAMPLE :

Given ๐‘“(๐‘ฅ) , take the following steps to find ๐‘“

โˆ’ 1

  1. Replace ๐‘“(๐‘ฅ) with y in the equation 2) ๐‘ฅ = 3 ๐‘ฆ โˆ’ 1

  2. Swap the x and y in the equation (they โ€œtrade placesโ€)

  3. Solve the equation for y (i.e., isolate the y ) 3) ๐‘ฅ + 1 = 3 ๐‘ฆ โ†’

๐‘ฅ+ 1

3

  1. Once y is isolated, replace it with ๐‘“

โˆ’ 1

โˆ’ 1

๐‘ฅ+ 1

3

or ๐‘“

โˆ’ 1

1

3

1

3

y

x

โˆ’ 1

๐‘ฅ+ 1

3

Notice how the graphs of ๐’‡(๐’™) and ๐’‡

โˆ’๐Ÿ

(๐’™) are symmetric (mirror images) across the diagonal line y = x , which is the result

of all the x - and y - values โ€œtrading placesโ€. Imagine each (๐‘ฅ, ๐‘ฆ) point as โ€œhoppingโ€ across the line y = x , becoming (๐‘ฆ, ๐‘ฅ).

โˆ’ 1

Check that these values work in both directions:

โˆ’ 1

๐‘ฅ+ 1

3