Operation on Functions | Addition, Subtraction, Multiplication and Division of Functions

Summary

In this educational video, the host, referred to as 'teacher,' explains the operations and functions of rational expressions. The script covers adding, subtracting, and dividing functions, specifically focusing on two functions, f(x) and g(x). Through step-by-step examples, the teacher demonstrates how to combine and simplify these functions, including function composition and evaluation at a specific point. The video aims to clarify these mathematical concepts for beginners, encouraging viewers to engage with the material and simplify their answers for clarity.

Takeaways

• 📚 The video is an educational tutorial focused on operations and functions in mathematics.
• 🔢 The script introduces two functions: f(x) = (x + 5) / (x - 7) and g(x) = 3 / (x - 7).
• ➕ The first operation discussed is the addition of functions f and g, resulting in f(x) + g(x) = (x + 8) / (x - 7).
• ➖ The second operation is the subtraction of g from f, leading to f(x) - g(x) = (x + 2) / (x - 7).
• 🔄 The third operation is the composition of functions, g(f(x)), which simplifies to 3(x + 5).
• 📉 The fourth operation involves evaluating g(f(x)) at a specific value, x = 2, resulting in g(f(2)) = 3/7.
• 📝 The importance of simplifying mathematical expressions is emphasized throughout the script.
• 👨‍🏫 The presenter, referred to as 'teacher', guides viewers through each step of the operations.
• 📹 The video script is repetitive, likely due to the nature of video recording, to ensure clarity.
• 📚 The script is designed for beginners in math classes, aiming to teach new techniques.
• 🔑 The video provides a step-by-step approach to combining and simplifying rational functions.
• 🌐 The video ends with a call to action for viewers to like, subscribe, and turn on notifications for more content.

Q & A

What is the main topic of the video?

The main topic of the video is the operation and function of mathematical expressions, specifically focusing on adding, subtracting, and dividing rational functions.

What are the two functions given in the video?

The two functions given are f(x) = (x + 5) / (x - 7) and g(x) = 3 / (x - 7).

What operation is performed in the first item of the video?

In the first item, the operation performed is the addition of the two given functions, f(x) and g(x).

How is the addition of f(x) and g(x) simplified in the video?

The addition is simplified by combining the numerators (x + 5 + 3) over the common denominator (x - 7), resulting in (x + 8) / (x - 7).

What is the result of f(x) + g(x)?

The result of f(x) + g(x) is (x + 8) / (x - 7).

What operation is performed in the second item of the video?

In the second item, the operation performed is the subtraction of g(x) from f(x).

How is the subtraction of g(x) from f(x) simplified in the video?

The subtraction is simplified by combining the numerators (x + 5 - 3) over the common denominator (x - 7), resulting in (x + 2) / (x - 7).

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

The result of f(x) - g(x) is (x + 2) / (x - 7).

What operation is performed in the third item of the video?

In the third item, the operation performed is the division of g(x) by f(x), which is also known as function composition.

How is the division of g(x) by f(x) simplified in the video?

The division is simplified by multiplying g(x) by the reciprocal of f(x), resulting in 3 * (x + 5) / (x - 7), which simplifies to 3x + 15.

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

The result of g(f(x)) is 3x + 15.

What is the value of g(f(2)) as calculated in the video?

The value of g(f(2)) is calculated by first finding f(2) = -7/5 and then applying g(x) to this result, which gives g(f(2)) = 3/7.

What technique is used to simplify the final answer in the video?

The technique used to simplify the final answer in the video is to cancel out common factors in the numerator and denominator and to evaluate the functions at specific values when necessary.

Outlines

00:00

📚 Introduction to Operations and Functions

The script begins with a casual greeting and introduction by the teacher, setting the stage for a lesson on operations and functions. The focus is on two given functions, f(x) = (x + 5) / (x - 7) and g(x) = 3 / (x - 7), which are rational functions. The teacher aims to demonstrate how to add, subtract, and divide these functions, providing a foundation for beginners in the subject. The description box is mentioned as a resource for further information. The first operation discussed is the addition of functions f and g, resulting in a simplified form of (x + 8) / (x - 7).

05:01

🔍 Subtracting Functions and Evaluating Compositions

This paragraph continues the mathematical theme by explaining how to subtract one function from another, specifically f(x) - g(x), which simplifies to (x + 2) / (x - 7). The teacher then introduces the concept of function composition, denoted as g(f(x)), and provides a step-by-step guide on how to perform this operation. The process involves dividing the functions and simplifying the result to 3(x + 5), showcasing the application of fraction division rules and the reciprocal concept.

10:02

📘 Evaluating Function Composition at a Specific Point

The final paragraph delves into evaluating the composition of functions g(f(x)) at a specific value, x = 2. The teacher demonstrates the process of substituting x with 2 in both functions f and g, and then performing the composition. The evaluation of f at x = 2 yields -7/5, and g at x = 2 yields -3/5. The composition g(f(2)) simplifies to 3/7 after canceling common factors and considering the signs of the numerators and denominators. The paragraph concludes with a reminder to subscribe to the channel for more educational content.

Keywords

💡Operation

In the context of the video, 'operation' refers to the mathematical procedures performed on functions, such as addition, subtraction, and division. It is integral to the theme as the video is focused on teaching how to manipulate and combine functions through various operations. For example, the script discusses 'f plus g of x' which is an operation of addition between two functions.

💡Function

A 'function' in mathematics is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. In the video, functions f(x) and g(x) are the primary subjects, with f(x) being 'x plus five over x minus seven' and g(x) being 'three over x minus seven'. The video explains how to perform operations on these functions.

💡Rational Functions

Rational functions are a type of function that can be represented as the ratio of two polynomials. In the script, both f(x) and g(x) are rational functions, as they are expressed as fractions where the numerator and denominator are polynomials. The video discusses how to add, subtract, and divide these types of functions.

💡Addition

Addition in the video script refers to the process of combining two functions to create a new function. It is demonstrated with 'f plus g of x', where the functions f(x) and g(x) are added together by combining their numerators over a common denominator.

💡Subtraction

Subtraction is another operation discussed in the video, which involves taking one function away from another. The script shows 'f minus g of x', where the function f(x) is subtracted from g(x), resulting in a new function with a simplified form.

💡Division

Division of functions is a process where one function is divided by another, as demonstrated in the script with 'g divided by f of x'. This involves multiplying by the reciprocal of the denominator, which is a key concept in the video.

💡Common Denominator

A 'common denominator' is a single denominator that is shared by all terms in a mathematical expression involving fractions. In the video, when adding or subtracting functions, the common denominator 'x minus seven' is used to combine the terms into a single fraction.

💡Numerator

The 'numerator' is the top part of a fraction, representing the number of parts being considered. In the video, when adding or subtracting functions, the numerators of the functions are combined over the common denominator to form a new function.

💡Denominator

The 'denominator' is the bottom part of a fraction, indicating the total number of equal parts into which the numerator is divided. In the script, the common denominator 'x minus seven' is used to simplify the addition and subtraction of functions.

💡Composition

In the context of the video, 'composition' refers to the process of applying one function to the result of another, denoted as g(f(x)). This is demonstrated in the script where the function g is applied to the result of function f, showcasing how to perform function composition.

💡Evaluation

Evaluation in the video script refers to the process of finding the value of a function at a specific point, such as when the script discusses 'g of f of 2'. This involves substituting the value of x into the functions f and g to find their respective values and then using those values in further calculations.

Highlights

Introduction to the topic of operation and functions in mathematics.

Explanation of two given functions, f(x) and g(x), with their respective expressions.

Demonstration of how to add two rational functions with similar denominators.

Simplification of the result after adding functions f(x) and g(x).

Subtraction of functions f(x) and g(x) with a step-by-step guide.

Simplification of the result after subtracting g(x) from f(x).

Introduction to the concept of function composition, g(f(x)).

Detailed process of dividing one function by another, showcasing function composition.

Cancellation technique used in simplifying the composition of functions.

Final simplified form of g(f(x)) after applying the division of fractions.

Evaluation of g(f(x)) at a specific value, x = 2, to demonstrate function application.

Separate evaluation of functions f and g at x = 2 for clarity.

Multiplication of the evaluated functions to find g(f(2)).

Final result of g(f(2)) presented with simplification.

Emphasis on the importance of simplifying mathematical answers for clarity.

Encouragement for new viewers to like, subscribe, and turn on notifications for channel updates.

Transcripts

hi guys hi guys hi guys hi guys hi guys

it's me teacher

in today's video

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

video we will talk about the operation

and function actually guys

on videos with regard to the operational

function

with regard to this kind of topic

in description box

so without further ado

let's do this topic so we have here

operation and functions and we are given

two different functions where we have f

of x which is equal to x plus five

over

x minus seven

and the other function is g of x is

equal to three over

x minus seven

and as you can see irrationals

rational functions

is to add

subtract

and divide these fractions or these

functions

and i hope number one new techniques

combine the beginner going into class in

given function so let's start with

number one we have f

plus g of x and target

is to add the two different functions so

let's have the f of x or in

aquino and atom f

plus g of x net n it will go like this

f

plus g

of x

and position

copy

x plus 5 is equal to

x

plus 5 over

x

minus seven

and then as you can see the operation

here is addition so we need to put here

plus

your g of x which is

three

over

x

minus

seven now initially

by guys as you can see their

denominators are similar

so i'm gonna bring it on these guys

but this time they are similar so all

you need to do

is to copy your common denominator which

is

x minus seven

and then add the numerators you have x

plus five plus

3

in the numerator we can combine 5 and 3

because the two numbers are considered

constants so we have here 5 plus 3 which

is 8. therefore

your f

plus g

of x

is simply

x

plus a

over

x minus seven this is the correct answer

for item

number one and then another thing guys

in problem

simplifying

final answer

because nothing

mathematical questions or answering math

problems i always simplify your answer

let's continue with item number two for

item number two

we have f minus g of x now as you can

see this one is subtracting or

subtraction of functions so we have f of

g

f minus g sorry

of x

we're in we need to copy your f of x

because

have

x plus 5

over

x minus 7

minus

your g of x which is 3

over

x

minus seven

since

uh this is minus what that is expected

is a good pattern they're different so

let's continue

same thing we need to do

since the denominators are common all

you need to do is to copy x minus seven

so your f

minus g

of x

is equal to

happier common denominator which is x

minus seven

and then subtract your numerators which

is x

plus five

minus

three now subtract privately

and it will give you

the f minus g

of x

be equal to

x

plus

2 because i minus 2

over

x

minus

7 and this is the answer

for item number 2 we're in

uh

an anatomy difference between

function f and function g

so i hope that one new method net end

suffers two

operations set in and now let's continue

with number three for number three we

have f sorry g

of f

of x oh sorry

g

divided by

f

of x so here

what we will do here is we will divide

the two functions paramount as g so

finally do i answer we have

g

of f

and g over f

or g divided f

of x that is composition

since that's of the end cg

what you will do is to copy

your

function g which is three

over

x

minus

seven next

all over

your function f which is

x

plus

five

over

x

minus

seven

what you will do is you would apply the

rules on how to divide fractions example

and guys what if you're asked to

divide one half

by one third what you will do is to copy

this one half

and then go in motion multiplication

and then get the reciprocal of one third

which is three over one so when you

multiply it away same thing in process

detail what you will do

is to copy your

numerator

three

over

over

x

minus seven

and then positive multiplication times

at the minus super high get the

reciprocal of this

this is x

minus seven

over

x plus

five for celebrating reciprocal guys uh

evaporating pressed to the numerators of

the laminator in that in numerator organ

denominator

in that in the denominator that w

numerator and then proceed to

multiplication but this time guys you

can cancel out x minus seven

because it has two x minus seven and as

a denominator x and x minus seven

so finally

your g of f

of x is simply

three times

that will give you x

plus

five and this is the answer

for item

number three so economists have attacked

events

okay now for item number four guys as

you can see you g of f10 is the same as

this one so on shortcut detail if my

encounter

all you need to do is to get this

you have

three

over

x plus five where in

vito your x is equal to two so we can

replace this x by two parameters

so this is three

over

two plus five now simplifying your

denominator this will give you jung g

how to do this

is to evaluate separately the function f

and function g

if the value of x is equal to so the

diagonal put in your f of x

is equal to

x plus five

over

x minus seven i can value the x naught

and a two so we can replace this by two

and then pythagorean variable x

determinant two it will become 2 plus 5

over

2 minus 7

by simplifying the top this will give

you negative

7 over 5

for f

now sir how about f of a g of two for g

of two

what will happen is that

copenhagen this is three over

x

minus seven

is equal to

but we have x beta 2 that is 3

over

2 minus 7.

this will give you this is negative

negative 3

over

five atoms

when you evaluate the function of g if

the value of x is two now

a m being a kind of secondary g

of f

over

your f which is negative

7 over 5.

uh copy your numerator

capillary

that is negative

3 over five

and then positive multiplication

wherein

this is negative

five over

seven

now as you can see maritime paradise

pipes are numerators denominator so we

can eliminate five and five and remember

negative times negative is positive so

we have here now the answer of g

of f

of two

be equal to

three

over

seven as you can see

paragraph supplementation is a good

method

it's a first weight way that

incompatibility number four and guys

uh i hope that is

operational

where by the way guys before you go into

so

so again guys if you're new to my

channel don't forget to like and

subscribe but hit that link

setting bel uploads again it's me that

you're going

by

bye