Operation on Functions/Teacher Espie
Summary
TLDRIn 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
🎓 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.
➕ 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.
➖ 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.
Mindmap
Keywords
💡Fundamental Operations
💡Function
💡Addition
💡Subtraction
💡Multiplication
💡Division
💡Sum
💡Difference
💡Quotient
💡Factoring
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
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