Operation on Functions | Addition, Subtraction, Multiplication and Division of Functions
Summary
TLDRIn 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
- π The video is an educational tutorial focused on operations and functions in mathematics.
- π’ The script introduces two functions: f(x) = (x + 5) / (x - 7) and g(x) = 3 / (x - 7).
- β The first operation discussed is the addition of functions f and g, resulting in f(x) + g(x) = (x + 8) / (x - 7).
- β The second operation is the subtraction of g from f, leading to f(x) - g(x) = (x + 2) / (x - 7).
- π The third operation is the composition of functions, g(f(x)), which simplifies to 3(x + 5).
- π The fourth operation involves evaluating g(f(x)) at a specific value, x = 2, resulting in g(f(2)) = 3/7.
- π The importance of simplifying mathematical expressions is emphasized throughout the script.
- π¨βπ« The presenter, referred to as 'teacher', guides viewers through each step of the operations.
- πΉ The video script is repetitive, likely due to the nature of video recording, to ensure clarity.
- π The script is designed for beginners in math classes, aiming to teach new techniques.
- π The video provides a step-by-step approach to combining and simplifying rational functions.
- π 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
π 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).
π 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.
π 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.
Mindmap
Keywords
π‘Operation
π‘Function
π‘Rational Functions
π‘Addition
π‘Subtraction
π‘Division
π‘Common Denominator
π‘Numerator
π‘Denominator
π‘Composition
π‘Evaluation
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
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