Composition of Functions - Grade 11 - General Mathematics

Summary

In this educational video, the teacher explains the concept of composite functions, denoted as f(g(x)) or f circle of G of X. The video uses f(x) = 3x - 5 and g(x) = x^2 + 2 to demonstrate how to evaluate f(g(2)) and g(f(-1)). The process involves substituting g(x) into f(x) for the former, and f(x) into g(x) for the latter, with calculations resulting in f(g(2)) = 13 and g(f(-1)) = 66. The teacher emphasizes the importance of understanding the order of functions in composite notation.

Takeaways

• 📚 The video is about composite functions or function composition.
• 🔢 The notation for composite functions is read as 'f circle of G of X' or 'F composite G of X', which is written as 'f(g(x))'.
• 💡 In 'f(g(x))', the function 'f' comes first, making 'g(x)' the input to function 'f'.
• 📐 The problem involves evaluating 'f(g(2))' and 'g(f(-1))' given two functions: f(x) = 3x - 5 and g(x) = x^2 + 2.
• 🧮 For 'f(g(2))', substitute 'x^2 + 2' for 'x' in the function 'f', resulting in '3(x^2 + 2) - 5'.
• 🔄 Simplify 'f(g(2))' to '3(2^2 + 2) - 5', which equals '3(4 + 2) - 5' or '3*6 - 5', resulting in 18 - 5 = 13.
• 🔄 For 'g(f(-1))', substitute '3x - 5' for 'x' in the function 'g', resulting in '(3x - 5)^2 + 2'.
• 🧮 Simplify 'g(f(-1))' to '(3*(-1) - 5)^2 + 2', which equals '(-3 - 5)^2 + 2' or '(-8)^2 + 2', resulting in 64 + 2 = 66.
• 📌 The video emphasizes that the order of functions in composite notation determines which function is the input.
• 🎯 The video aims to teach viewers how to evaluate composite functions and encourages them to like, subscribe, and hit the bell for updates.

Q & A

What is a composite function?

A composite function is a function that is formed by applying one function to the result of another function. It is denoted as f(g(x)) or f circle of g of x.

How is f(g(x)) different from g(f(x))?

f(g(x)) means that the function g(x) is used as the input for the function f, while g(f(x)) means that the function f(x) is used as the input for the function g. The order of the functions matters.

What are the given functions in the transcript?

The given functions are f(x) = 3x - 5 and g(x) = x^2 + 2.

How is f(g(x)) evaluated when x is replaced by 2?

To evaluate f(g(2)), you first calculate g(2) which is 2^2 + 2 = 6, then substitute this into f(x) to get f(6) = 3*6 - 5 = 18 - 5 = 13.

What is the result of f(g(2))?

The result of f(g(2)) is 13.

How is g(f(x)) evaluated when x is replaced by -1?

To evaluate g(f(-1)), you first calculate f(-1) which is 3*(-1) - 5 = -8, then substitute this into g(x) to get g(-8) = (-8)^2 + 2 = 64 + 2 = 66.

What is the result of g(f(-1))?

The result of g(f(-1)) is 66.

What does the notation f circle of g of x mean?

The notation f circle of g of x means the composition of function g with function f, which is read as 'f composite g of x'.

Why is it important to follow the order of functions when evaluating composite functions?

The order of functions is important because it determines which function's output becomes the input for the other function. Changing the order can lead to different results.

What is the significance of the variable x being replaced by another function in composite functions?

In composite functions, replacing the variable x with another function allows for the application of one function's output to another function, which is the core concept of function composition.

How can one avoid confusion between the notations f(g(x)) and g(f(x))?

To avoid confusion, remember that the function that comes first in the notation is the outer function, and the one that comes second is the inner function whose result is used as the input for the outer function.

Outlines

00:00

📚 Introduction to Composite Functions

The speaker, Teacher, introduces the concept of composite functions, which are functions within functions. The notation f(g(x)) or f ∘ g(x) is explained, where 'f' is the outer function and 'g' is the inner function. The speaker emphasizes that the order of functions indicates the order of operation, with 'f' being applied after 'g'. The video then presents a problem involving two functions: f(x) = 3x - 5 and g(x) = x^2 + 2. The task is to evaluate f(g(2)) and g(f(-1)). The speaker begins by evaluating f(g(x)) by substituting g(x) into f(x), resulting in 3(x^2 + 2) - 5. The substitution is then made with x = 2, leading to 3(2^2 + 2) - 5, which simplifies to 3(4 + 2) - 5 = 3(6) - 5 = 18 - 5 = 13. The speaker clarifies that in composite functions, the function that appears first in the notation is applied last to the input.

05:02

🔍 Evaluating Composite Functions: Part 2

Continuing from the previous explanation, the speaker now evaluates g(f(-1)). This involves substituting f(x) into g(x), resulting in (3x - 5)^2 + 2. The substitution is made with x = -1, leading to (3(-1) - 5)^2 + 2, which simplifies to (-3 - 5)^2 + 2 = (-8)^2 + 2 = 64 + 2 = 66. The speaker concludes by summarizing the results: f(g(2)) = 13 and g(f(-1)) = 66. The video aims to educate viewers on how to evaluate composite functions and encourages new subscribers to like, subscribe, and enable notifications for updates. The speaker signs off with their name, Turgon.

Keywords

💡composite function

A composite function is a function that is formed by applying one function to the result of another. It is denoted as f(g(x)) or f ∘ g(x), where 'f' and 'g' are functions and 'x' is the input variable. In the video, the concept is central as it illustrates how to evaluate composite functions by substituting one function's output into another, as shown in the examples of f(g(2)) and g(f(-1)).

💡function composition

Function composition is the process of combining two functions to produce a third function. It is a fundamental concept in mathematics, particularly in calculus and functional programming. The video script explains this by showing how to calculate f(g(x)) by first evaluating g(x) and then substituting that result into f(x).

💡notation

In mathematics, notation is a system of symbols used to represent operations, quantities, or relations. The script discusses the notation for composite functions, f(g(x)), emphasizing how it indicates the order in which functions are applied. The notation is crucial for understanding the sequence of operations in composite functions.

💡input function

The input function in a composite function is the one that is evaluated first and whose result is then used as the input for the second function. In the context of the video, g(x) is the input function when calculating f(g(x)), as it is evaluated before being substituted into f(x).

💡evaluation

Evaluation in the context of composite functions means calculating the value of the function for a given input. The video provides a step-by-step guide on how to evaluate composite functions, such as finding f(g(2)) and g(f(-1)), by substituting the inner function's output into the outer function.

💡variable substitution

Variable substitution is a technique used in mathematics where a variable is replaced with an expression or a value. In the video, variable substitution is used to solve composite functions by replacing 'x' in f(x) with the result of g(x) or vice versa.

💡function f

Function f is one of the two functions used in the video to demonstrate composite functions. It is defined as f(x) = 3x - 5. The video uses this function to show how to apply function composition by substituting the output of g(x) into f(x).

💡function g

Function g is the other function used in the demonstration, defined as g(x) = x^2 + 2. The script explains how to evaluate composite functions involving g, such as when g is the input function for f.

💡simplification

Simplification in mathematics refers to the process of making an expression easier to understand or calculate. The video script includes simplification steps when evaluating composite functions, such as simplifying 3 * (2^2 + 2) - 5 to find f(g(2)).

💡operation order

Operation order, or the order of operations, is a set of rules that dictate which operations to perform first in a given mathematical expression. The video emphasizes the importance of operation order in evaluating composite functions, ensuring that functions are composed correctly.

💡example problems

Example problems are used in educational content to illustrate how to apply concepts or solve problems. The video provides example problems involving composite functions, such as evaluating f(g(2)) and g(f(-1)), to help viewers understand the process of function composition.

Highlights

Introduction to composite functions

Notation for composite functions explained

Understanding the order of functions in composite notation

Given functions f(x) = 3x - 5 and g(x) = x^2 + 2

Evaluating f(g(2)) step by step

Substituting g(x) into f(x) for f(g(2))

Simplifying the expression for f(g(2))

Final calculation for f(g(2)) equals 13

Clarification on the difference between f(g(x)) and g(f(x))

Evaluating g(f(-1)) step by step

Substituting f(x) into g(x) for g(f(-1))

Simplifying the expression for g(f(-1))

Final calculation for g(f(-1)) equals 66

Emphasis on the importance of function order in composite functions

Encouragement for new viewers to subscribe and engage with the channel

Transcripts

hi guys it's me teacher going in today's

video we will talk about the composite

function or composition of function

so we have here this notation

this is read as f

circle of G of X or

F composite G of X and that is equal to

this notation f g of X we're in in this

function since the variable F or the

function f comes first before G

G will be the input or the input

function

with respect to the function f so

without further Ado let's do this topic

so let's have this problem

we are given two different functions f

of x is equal to 3x minus 5 and the

other is 3x at 3 of x g of x

is equal to x squared plus two and right

now we are asked to

evaluate this one we have f

of G of 2 and the other is G of f of

negative one let's start with number one

in this case RF of G of 2 here

first we will try to evaluate

f

of G

of X that is the same as that

okay now what will happen here is that

in this case guys

we will move this one

since

your function f

comes first before G

we will plug in

this x square plus 2

to replace the variable X in

3x minus five so it goes like this three

copy three

then your X this x will be replaced by x

square

times x square

plus 2

minus 5. so here what is the reason Kai

that is f

look at this one your gfx is inside the

parenthesis this will be the input

so we will replace the variable x 3

times x square

plus 2 minus 5. so what will happen

after that is as you can see

is that we will replace this variable X

by 2 and it will become

3

times

2 square

plus 2 minus 5. so what will happen is

that you need to simplify this to become

3

times 2 squares four

plus two then we have minus five

simplify

three

times four plus two which is equal to

six

and then minus 5.

so this is three times six is eighteen

minus five therefore

your f

of G

of 2 is equal to 15 minus 18 minus 5

which is equal to 13. guys don't be

confused about these two different

notations if the variable F comes first

meaning your input is

the function G

if the function G comes first

your function f will be the input okay

let's continue

here number two we have G of f of

negative one

so it will become

G

off

f

of X it goes like this meaning

we will use this and this 3x minus 5

will be replaced or we will be using

this to substitute for the value

variable X and it will become like this

x squared that is x squared by copy this

one

3x minus 5

Square

and then don't forget plus two sure

X squaring

now after doing that remember that you

are asked to evaluate negative one

we will replace this x by negative one

it will become

3

times negative one

minus 5 square plus two

so simplify that into three times

negative 1 is negative

three

minus five

then square plus 2. so simplify that

until your negative three minus five is

negative eight

Square

plus 2 and negative 8

squared is not negative 64. right the

answer is 60 4 and then plus 2.

therefore

your G

can you say this here G

of f

of negative 1

is equal to 64 plus 1 and that is 66.

and this is the answer guys

so I hope guys learned something from

this video on how to

evaluate composite functions so if

you're new to my channel don't forget to

like And subscribe button hit the Bell

button for you to be updated starting

latest uploads again it's me to turgon

my name is