Composite Function | General Mathematics @MathTeacherGon
Summary
TLDRIn this educational video, the host, 'Teacher,' delves into the concept of function composition, a topic distinct from operational function evaluation due to its reliance on substitution. The video explains the process using two functions, f(x) = x^2 + 5x + 6 and g(x) = x + 2. Through step-by-step examples, the host demonstrates how to compute f(g(x)), g(f(x)), and specific values like f(g(4)), guiding viewers through the substitution and simplification process. The tutorial is designed to help viewers understand and perform function composition effectively.
Takeaways
- đ The video discusses the concept of function composition, which is a method of combining two functions to create a new function.
- đą The functions used as examples are f(x) = x^2 + 5x + 6 and g(x) = x + 2, where f and g are the primary functions being composed.
- đŻ The video explains how to find the composition of functions, specifically f(g(x)) and g(f(x)), by substituting one function into another.
- đ The process involves replacing the variable x in the function with the entire expression of the other function, demonstrating the substitution method.
- 𧟠The video provides a step-by-step calculation for f(g(x)) by substituting g(x) = x + 2 into f(x), resulting in a simplified expression.
- đ For g(f(x)), the process is reversed, with f(x) substituted into g(x), leading to another simplified expression.
- đ The video emphasizes the importance of simplifying expressions by combining like terms and using algebraic properties.
- đ An example with a specific value (f(g(4))) is used to demonstrate how to evaluate the composition of functions at a given point.
- đšâđ« The presenter, Teacher, encourages viewers to practice and understand the concept of function composition, suggesting it's different from evaluating operational functions.
- đą The video concludes with a call to action for viewers to like, subscribe, and stay updated with the channel for more educational content.
Q & A
What is the main topic of the video?
-The main topic of the video is the composition of functions.
What are the two functions provided in the video?
-The two functions provided are f(x) = x^2 + 5x + 6 and g(x) = x + 2.
What does the notation 'f â g' represent?
-The notation 'f â g' represents the composition of the function f with the function g, which means applying g first and then f to the result.
How is the composition of functions different from evaluating operational functions?
-The composition of functions involves a lot of substitution, where one function is used as the input for another, which is different from evaluating operational functions where you directly substitute the variable.
What is the result of the composition f(g(x))?
-The result of the composition f(g(x)) is f(x + 2), which simplifies to x^2 + 9x + 20.
What is the result of the composition g(f(x))?
-The result of the composition g(f(x)) is g(x^2 + 5x + 6), which simplifies to x^2 + 5x + 8.
What is the value of f(g(4))?
-The value of f(g(4)) is 72, after evaluating g(4) to get 6 and then substituting 6 into f(x) to get f(6).
What is the process for evaluating a composition of functions?
-The process involves substituting the inner function into the outer function, simplifying the expression, and then evaluating if a specific input is given.
What is the significance of the distributive property in the context of the video?
-The distributive property is significant as it is used to expand and simplify the expressions when substituting and evaluating the compositions of functions.
How does the video script guide viewers through the process of function composition?
-The video script guides viewers step-by-step through the process of function composition by providing clear examples and explanations of how to substitute and simplify expressions.
Outlines
đ Introduction to Composition of Functions
This paragraph introduces the concept of function composition, a topic that is distinct from evaluating operational functions due to the extensive use of substitution. The teacher promises to explain the process thoroughly. The functions given are f(x) = x^2 + 5x + 6 and g(x) = x + 2. The composition f(g(x)) is explained, where g(x) is substituted into f(x), resulting in f(g(x)) = f(x + 2). The process involves substituting x with (x + 2) in the function f(x), leading to a new expression which is then simplified using algebraic methods. The final simplified form of f(g(x)) is x^2 + 9x + 20.
đą Function Composition: g(f(x)) and Specific Value Example
The second paragraph continues the discussion on function composition but focuses on g(f(x)) and includes an example with a specific value. It starts with explaining g(f(x)) by substituting f(x) into g(x), which results in g(x^2 + 5x + 6). The substitution leads to a new expression that simplifies to x^2 + 5x + 8. The paragraph then presents an example where the composition of functions is evaluated at a specific input, x = 4. The process involves first evaluating g(4), then substituting this result into f(x) to get f(g(4)). The final result of f(g(4)) is calculated to be 72, showcasing a step-by-step approach to evaluating function compositions at specific values.
đ Conclusion and Channel Engagement
The final paragraph is a brief conclusion where the teacher signs off with a goodbye. It also serves as a call to action for viewers, encouraging them to like and subscribe to the channel for updates on the latest uploads. The teacher, identified as 'turgon', uses this opportunity to remind viewers of the channel's purpose and to engage with the audience.
Mindmap
Keywords
đĄComposition of Functions
đĄFunction
đĄSubstitution
đĄSquaring
đĄDistributive Property
đĄLike Terms
đĄEvaluation
đĄOperational Functions
đĄBinomial
đĄDomain
Highlights
Introduction to the composition of functions and its difference from evaluating operational functions.
Explanation of the composition method or pattern of g(f(x)) and f(g(x)).
Definition of the given functions f(x) = x^2 + 5x + 6 and g(x) = x + 2.
Step-by-step process to find f(g(x)) by substituting g(x) into f(x).
Simplification of the expression f(g(x)) by expanding and combining like terms.
Final expression for f(g(x)) which is x^2 + 9x + 20.
Process to find g(f(x)) by substituting f(x) into g(x).
Simplification of g(f(x)) resulting in the expression x^2 + 5x + 8.
Explanation of how to handle specific values in composition, using f(g(4)) as an example.
Evaluation of g(4) to find the input for f(x).
Substitution of the value from g(4) into f(x) to find f(g(4)).
Final calculation of f(g(4)) which results in the value 72.
Emphasis on the importance of understanding the composition of functions for those new to the topic.
Encouragement for viewers to like and subscribe to the channel for updates on similar educational content.
Closing remarks by the teacher, signing off with a friendly tone.
Transcripts
hi guys hi guys hi guys hi guys hi guys
it's me teacher
in this video
hi guys it's me teacher going in our
today's video we will talk about the
composition of function
so
this topic
but i will try my best to deliver this
one namaste
because in compositional function this
one is quite different from evaluating
an operational functions because she
will do a lot of
substitution here so without further ado
let's do this topic guys so we have here
the home position of function in general
formation method or pattern atom
of g of x so these two guys i will
explain first
f
g so get the information
on your i
g
of
we have here the given functions f of x
is equal to x squared plus five x plus
six
and then g of x is equal to x plus two
we are given two functions and then we
need to try these three examples
we have here f compose of g
of x so facility input
f
so
we have f
of g
time
f
of g
of x
so copy that into
f
of g
of x where in again
your function g is the input so what we
will do here is we will
get this value because g of x is equal
to x plus 2 since they are equal we can
replace it by x plus 2 so it will become
f
of
f
of x plus 2
we will use x plus 2 as our input
satin expression okay as a function f of
function f this is your function f atom
again input and icg or x plus 2 it will
become x squared
plus 5 x plus 6 again this is your g of
g of x y
so what we will do is we'll replace all
the x variable here and it will become
x plus 2
squared so considering your x naught n
squared negative x plus 2 squared
tables plus 5 times x plus two
plus six
and ito
we can simplify this
by
square binomial that will express the
squared data is the same as
x squared
plus four x
plus four and shortcut and now guys
perpendicular method theta determined by
distributive property it will become
plus five x
equal five times 2 plus 10
plus 6
okay
now
for this time domain guys combine like
terms you have x squared
beta you have plus 9x because 4x sub 5x
for the constant we have 4 plus 10 that
is
14 plus 6 that will give you
the answer of 20. if you said
i'm adding
f
of x plus 2 is equal to x squared plus 9
x plus 20 or eternity
we have
f
of g
of x so again
we have x squared
plus
nine x plus one e okay so brighten up
now let's move on with item number two
for item number two we are given g of f
of x
okay basically information here if you
have g
of f
of x
that is this one i said g that's another
parenthesis c function f
now
since your adding input is f of x
it will become
g
of x squared
plus five x
plus six eternal initial
plus 5x plus six
is equal to and copenhagen function g
x plus two hang on again this is your
uh this is your x and you know the input
and
so we will replace x by x squared plus
five x plus six
that will become x squared
plus five x
plus six and then plus
two and simplifying this your g of x
squared plus five x
plus six
is equal to
x squared
plus five x
plus this is eight
and capacition it up so we will write
here our answer which is x squared
plus five x
plus e
for number one number two second one
as a basic fine given function
i'm again
lots of exponents you do not know
how to perform operations
so let's continue let's try having
item number three writer number three
this one is quite different because
our input here we have a specific value
for our input which is four we're all
going for
we have
f
compose g or f circle g of four
and that is the same as
f
and this log means
of
4 okay
so what we will do here is we will
try to announce try to evaluate g
function g so
g
of 4
is equal to
x plus 2. this is
and your g of four
is simply
four plus two
or in other words this is
g of four
is equal to six
methanol in geo format
okay so
cf
of g
again
uh in g of four negative divided by six
because they are equal top was
contaminated function at the
x squared plus five x
plus six atom in six metal gallium satin
geo four we will replace excess here by
six it will become
six x six squared
plus five times six
plus six
okay so
guys
we have here 36
plus 30 plus 6. but in advan your f of 6
is equal to 36 plus
30 this is 66 plus 6 this will give you
72. in this opinion guys if f of 6 is
equal to 7 2 your value num
f of g
of 4 is equal to 72.
so i'm going to show you guys on how to
do compositional functions so
on how to do composition of functions so
again if you are new to my channel don't
forget to like and subscribe but hit
like button for you to be updated at the
latest uploads again it's me turgon
bye
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