Operation on Functions/Teacher Espie

Summary

In this educational video, Teacher SV explains the four fundamental operations on functions: addition, subtraction, multiplication, and division. The video breaks down each operation using clear examples, starting with the basics of adding and subtracting functions, moving on to multiplying polynomials, and concluding with division. Teacher SV also discusses important rules, such as handling undefined values when dividing by zero. The video includes step-by-step explanations of factoring, using the FOIL method, and simplifying answers, making it easy for students to understand. Viewers are encouraged to subscribe for more tutorials.

Takeaways

• 📚 The video focuses on teaching the four fundamental operations on functions: addition, subtraction, multiplication, and division.
• ➕ The sum of two functions is defined as f(x) + g(x), similar to adding integers.
• ➖ The difference between two functions is defined as f(x) - g(x), similar to subtracting integers.
• ✖️ The product of two functions is defined as f(x) * g(x).
• ➗ The quotient of two functions is defined as f(x) / g(x), with the condition that g(x) must not be zero to avoid an undefined result.
• 🔢 Example functions f(x) = x² - 8x + 16 and g(x) = x - 4 are used to demonstrate the operations.
• 🧮 The instructor demonstrates the process of adding functions, following integer addition rules for combining terms.
• 📝 Subtracting functions involves changing the signs of the subtrahend and then proceeding with addition rules.
• 🔀 Multiplication of functions is shown using a polynomial multiplication technique.
• ❌ Division is explained by factoring and canceling common terms between the numerator and denominator.

Q & A

What are the four fundamental operations of functions?

The four fundamental operations of functions are addition, subtraction, multiplication, and division.

How is the sum of two functions, f(x) and g(x), defined?

The sum of two functions, f(x) and g(x), is defined as f(x) + g(x).

How do you calculate the difference of two functions?

The difference of two functions, f(x) and g(x), is calculated as f(x) - g(x).

What is the product of two functions, f(x) and g(x)?

The product of two functions, f(x) and g(x), is defined as f(x) * g(x).

How do you define the quotient of two functions?

The quotient of two functions, f(x) and g(x), is defined as f(x) / g(x), where g(x) is not equal to zero.

Why is division by zero undefined?

Division by zero is undefined because it leads to an indeterminate form. For example, dividing any number by zero does not yield a valid number.

In the given example, f(x) = x^2 - 8x + 16 and g(x) = x - 4, how do you compute the sum of f(x) and g(x)?

To compute the sum, add the functions: (x^2 - 8x + 16) + (x - 4). The result is x^2 - 7x + 12.

What method is used to verify the result of factoring in this lesson?

The FOIL method (First, Outer, Inner, Last) is used to verify the result of factoring.

How do you handle subtraction of two functions in the example provided?

To handle subtraction, change the signs of the second function (subtrahend) and then proceed with addition. For example, subtracting g(x) from f(x) becomes: (x^2 - 8x + 16) - (x - 4), which simplifies to x^2 - 9x + 20.

How is the product of the functions f(x) and g(x) computed in the example?

To compute the product, multiply the polynomials: (x^2 - 8x + 16) * (x - 4). The result is x^3 - 12x^2 + 48x - 64.

How do you perform division of two functions in the example?

For division, factor both f(x) and g(x). In this case, f(x) = (x - 4)(x - 4) and g(x) = (x - 4). Cancel the common factor (x - 4), leaving the result as x - 4.

Outlines

00:00

🎓 Introduction to Fundamental Operations on Functions

In this introduction, Teacher SV welcomes viewers and outlines the lesson topic: fundamental operations on functions. The four main operations—addition, subtraction, multiplication, and division—are briefly introduced using examples with basic numbers (e.g., 8 + 9, 18 - 9). The terms 'sum,' 'difference,' 'product,' and 'quotient' are explained within the context of functions, with emphasis on division and the importance of ensuring that the denominator is not zero to avoid an undefined result.

05:01

➕ Addition of Functions Explained with Example

The focus here is on adding two functions, f(x) and g(x), using specific examples. Teacher SV shows the process of adding two polynomial functions, f(x) = x² - 8x + 16 and g(x) = x - 4, by following the rules of adding integers, such as handling like and unlike signs. The steps for simplifying the result and factoring are also demonstrated using the FOIL method to check the answer, reinforcing how to properly combine functions through addition.

10:03

➖ Subtracting Functions: Key Steps and Example

In this section, subtraction of functions is covered, using the same example functions, f(x) = x² - 8x + 16 and g(x) = x - 4. Teacher SV explains the importance of changing the signs of the subtrahend before applying the rule of addition. The steps involve adjusting signs, combining like terms, and checking if the expression can be factored. The factoring process is demonstrated again, ensuring that students know how to simplify the expression correctly.

✖️ Multiplication of Functions: Polynomial Multiplication

This paragraph covers the multiplication of functions, f(x) = x² - 8x + 16 and g(x) = x - 4. Teacher SV introduces polynomial multiplication, explaining the importance of multiplying each term in one function by each term in the other. The example shows the resulting terms, such as x³, and how to combine them after multiplication. The concept of arranging terms by degree is highlighted, and the final product is simplified.

➗ Division of Functions: Simplifying Fractions

The division of functions is explained using the example f(x) = x² - 8x + 16 divided by g(x) = x - 4. Teacher SV demonstrates factoring to simplify the functions before division, canceling out common factors in the numerator and denominator. The importance of ensuring that the denominator is not zero is reinforced, and the final simplified result is presented. The paragraph concludes with a step-by-step explanation of the division process for functions.

✅ Conclusion: Recap of Function Operations and Farewell

In this closing section, Teacher SV summarizes the key points covered in the lesson about the four fundamental operations on functions. Viewers are encouraged to subscribe, like, and hit the notification bell to stay updated on future videos. Teacher SV also invites viewers to request topics for future discussions, offering a personal touch. The video ends with a friendly farewell.

Keywords

💡Fundamental Operations

The four fundamental operations of mathematics—addition, subtraction, multiplication, and division—are central to the video. The speaker explains how these operations apply to functions, such as how to sum or subtract two functions like f(x) and g(x). For example, the sum of f(x) = x² - 8x + 16 and g(x) = x - 4 is demonstrated by adding their corresponding terms.

💡Function

A function, often denoted as f(x) or g(x), represents a mathematical relationship between an input and an output. In the video, the teacher uses f(x) and g(x) to demonstrate how to apply mathematical operations like addition and subtraction to functions. An example function is f(x) = x² - 8x + 16, which is used to show how to compute the sum or product of two functions.

💡Addition

Addition is one of the fundamental operations discussed in the video. It is applied to functions by adding their corresponding terms, like adding f(x) and g(x). The speaker gives an example where adding f(x) = x² - 8x + 16 and g(x) = x - 4 results in x² - 7x + 12. This operation helps learners understand how to combine functions.

💡Subtraction

Subtraction is another core operation explained in the video. It involves subtracting one function from another by changing the sign of the subtracted function's terms before performing the operation. For example, subtracting g(x) from f(x) requires changing the signs of g(x) before combining the terms of f(x) and g(x).

💡Multiplication

The video explains how to multiply functions by multiplying their corresponding terms. The example provided is multiplying f(x) = x² - 8x + 16 and g(x) = x - 4, which results in a polynomial expression. The speaker also touches on polynomial multiplication, demonstrating how to apply distributive property.

💡Division

Division is the last of the four operations discussed, where one function is divided by another. The video provides an example of dividing f(x) = x² - 8x + 16 by g(x) = x - 4, where factoring is used to simplify the expression. The teacher emphasizes the importance of ensuring that the denominator is not zero to avoid undefined results.

💡Sum

The 'sum' refers to the result of adding two functions. The video shows how the sum of f(x) and g(x) is calculated by adding their respective terms. For example, when adding f(x) = x² - 8x + 16 and g(x) = x - 4, the sum is x² - 7x + 12. This concept helps reinforce the rules of combining like terms.

💡Difference

The 'difference' is the result of subtracting one function from another. The video explains how to compute the difference by changing the signs of the terms in the second function before combining. For instance, subtracting g(x) = x - 4 from f(x) = x² - 8x + 16 results in x² - 9x + 20.

💡Quotient

The quotient is the result of dividing one function by another. The teacher shows how to divide f(x) by g(x) and simplify the expression by factoring. In the example, f(x) = x² - 8x + 16 is divided by g(x) = x - 4, and factoring simplifies the expression to x - 4. This teaches the concept of reducing rational expressions.

💡Factoring

Factoring is a technique used to simplify expressions, especially in multiplication and division. In the video, the teacher shows how to factor polynomials like x² - 8x + 16 to simplify operations such as division. Factoring plays a crucial role in reducing the quotient of two functions, allowing for easier manipulation of algebraic expressions.

Highlights

Introduction to the topic of operations on functions.

Explanation of the four fundamental operations: addition, subtraction, multiplication, and division.

Definition of the sum of two functions: f(x) + g(x).

Demonstration of subtraction of functions: f(x) - g(x).

Discussion on multiplication of functions: f(x) * g(x).

Introduction to division of functions with the condition that g(x) ≠ 0 to avoid undefined results.

Explanation of why division by zero is undefined and how to check for it.

Example of addition of functions using f(x) = x² - 8x + 16 and g(x) = x - 4.

Step-by-step guide on how to simplify the sum of functions using factoring and the FOIL method.

Example of subtracting functions with the same f(x) and g(x).

Clarification on changing the sign of the subtrahend before applying the addition rule.

Explanation of factoring in the subtraction example and validation using the FOIL method.

Example of multiplying functions using f(x) = x² - 8x + 16 and g(x) = x - 4.

Step-by-step multiplication process of a polynomial by a binomial.

Example of division of functions, including factoring and canceling common terms to simplify the result.

Transcripts

[Music]

hi guys this is teacher sv

and welcome to my channel so today

i will be teaching you about operation

on function so before i'm going to

discuss let us review

what are the four fundamental operation

of a function

so if you're ready so please keep

watching until the end of this

[Music]

video

[Music]

[Music]

so when we are talking about the four

fundamental operations guys

these are addition subtraction

multiplication and division

addition for example

8 plus 9 will give you 17

and what do you call the answer here and

that is the sum

so therefore the sum of

f of x and g of x is defined

as f of x plus

g of x okay so this is the formula

for the sum and if you are going to

subtract

18 minus 9 will give you 7

and the answer for subtraction is

the difference so therefore the

difference of f

of x and g of x is defined

as f of x

minus g of x

okay and if you're going to multiply

4 times 3 will give you 12 and the

answer

is product and therefore

the product of f of x and g of x

is defined as f

of x and g

of x and if you're going to divide

16 and 2 will give you 8

the answer is quotient the quotient

of f and x and g of x

is defined as f

of x divided by

g of x such that

g of x is not equal to zero

why what happened if your denominator is

zero

that the the value is undefined

all variables all numbers see to it that

the denominator is not equal to zero why

because if you're going to prove eight

over zero

is equal to zero an answer is zero

zero if you are going to check your

answer zero times zero will give you

zero and the answer is eight

eight eight times zero is equal to eight

so that's why

all whole numbers all variables and if

the denominator is zero

that is undefined you cannot define it

okay so into the

four fundamental operations of function

if we are going to use

this in your equation so when you are

talking about

function f of x and g of x so for

example

if i will be giving you an example f of

x is equal to

x squared minus 8x

plus 16 and the value of g

of x is equal to x minus 4.

so it'll be given

fundamental operation so first we have

the sum which is f of x

and g of x so

we know that our value of your f of x is

equal to

x squared minus eight x

plus 16 and x minus four

so

now x minus 4

and we are going to follow the rule in

adding integers

when you are at when you are adding

integers

from my previous video unlike sine you

are going to subtract and copy the sign

of the bigger number

like sign you are going to add so 16

minus 4 will give you positive 12.

so subtract the the unlike sine positive

and the value

of 16 is 16 is higher than

4 so you have to follow the sine of 16

and then negative 8 and positive 1

subtract and copy the sign of the bigger

number will give you

negative 7x

and uh we have x squared bring down the

quotient x squared so it took a lot but

before if you're going to simplify

that simplify

now if you're going to add is equal to 7

can you give me the factors of 12. the

factors of 12 are

four and three so we are going to say

factor

multiply with that is equal to twelve

four and three

six and two twelve and one

factor number

so if you are going to simplify this

using factoring

then factor is x

and factoring ix factored in 12

is 4 and 3 and if the

sign of the middle term is negative

then all of this is negative

how are you going to check your answer

using the foil method

so if you're going to use the foil

method x times

x will give you x squared outer term

x times negative 3 will give you

negative 3x

negative 4 times x will give you

negative 4x so

parenthesis sign then you are going to

add so negative 4 and negative 3 will

give you

negative 7x and 4 times negative 3 will

give you

positive 12 and that is correct so see

to it that you are going

to simplify your answer to lowest term

and using the factory

but if it is not factorable then that's

it that is your final answer

okay so another one same example

we are going to use same example but we

need to subtract

it's just subtracted so same equation

f of x minus g of x okay

so so letter b letter b

is f of x minus

g of x it's a subtract so am i giving x

squared

minus eight x plus sixteen

and x minus four when you are

subtracting a number

see to it that you are going to change

the sign of the subtrahend before you

proceed to

the rule of addition this positive will

become negative

and negative will become positive tapos

must proceed to the rule of addition

positive 16 and positive 4 will give you

positive 20 negative 8

and negative 1 will give you a negative

9x

bring down x squared and then check

if it is factorable so i know i'm

factoring 20 if you are going to add is

equal to nine

so i'm factoring 20 i five and four

20 and one okay so

10 and two out of the three f factors

and factor num 20 if you're going to add

is equal to 9

so that is 5 and 4. so this is

simplified

the first term which is x squared so x

and x

factor of 20 so am i gigging five plus

my digging four

and the sign that the middle term is

negative so negative negative

therefore x minus five

x minus 4 then check again using the

foil method

and if the answer is equal to this your

answer is correct so this is now

our final answer okay so this is for

the subtraction but how about for letter

c we have

multiplication so x squared minus eight

x plus 16 times

x minus four we will be using a

multiplication of a polynomial using a

long

1368 times 218.

so if i say it's a decision multiplicand

this is a multiplier

so positive 16 times negative 4 will

give you negative

64. negative 4 times negative 8x will

give you

positive 32x negative 4

times x squared will give you negative

four

x squared so we're done already with

negative four

what happens so x times positive sixteen

positive sixteen x and then

negative eight x times x my gig negative

8x squared

x squared times x will give you x cubed

so now bring down negative 64.

32 plus 16 will give you positive 48x

negative 4 and negative 8 will give you

negative 12 x squared so

and then bring down x cube so therefore

the product of

is equal to x squared negative 12 x

squared

plus 48 x and negative 64.

okay so that is for multiplication how

about

if you are going to use division

so for the division an f of exponential

so for letter d and f of x

i x squared negative 8

x plus 16 divided by

g of x x which is x minus 4.

so let us check or nothing guys

factor in

to eight that is four and four so

therefore

by getting x minus four and then

x minus four then bring down

x minus four so if you're going to

simplify that

you can cancel a numerator and a

denominator

so cancel x minus four x minus four

and that is equal to x minus

four so this is now the final answer so

this is now the process

on how are you going to use the four

fundamental operations

using a function okay so

uh i hope guys that you learned

something today a

don't forget to subscribe like and

hit the notification bell so that you

will be updated for whatever

videos i'm going to upload and by the

way guys you can request

the topic that you are going to discuss

and have a great day everyone bye