Composition of Functions - Grade 11 - General Mathematics
Summary
TLDRIn this educational video, the teacher explains the concept of composite functions, denoted as f(g(x)) or f circle of G of X. The video uses f(x) = 3x - 5 and g(x) = x^2 + 2 to demonstrate how to evaluate f(g(2)) and g(f(-1)). The process involves substituting g(x) into f(x) for the former, and f(x) into g(x) for the latter, with calculations resulting in f(g(2)) = 13 and g(f(-1)) = 66. The teacher emphasizes the importance of understanding the order of functions in composite notation.
Takeaways
- đ The video is about composite functions or function composition.
- đą The notation for composite functions is read as 'f circle of G of X' or 'F composite G of X', which is written as 'f(g(x))'.
- đĄ In 'f(g(x))', the function 'f' comes first, making 'g(x)' the input to function 'f'.
- đ The problem involves evaluating 'f(g(2))' and 'g(f(-1))' given two functions: f(x) = 3x - 5 and g(x) = x^2 + 2.
- 𧟠For 'f(g(2))', substitute 'x^2 + 2' for 'x' in the function 'f', resulting in '3(x^2 + 2) - 5'.
- đ Simplify 'f(g(2))' to '3(2^2 + 2) - 5', which equals '3(4 + 2) - 5' or '3*6 - 5', resulting in 18 - 5 = 13.
- đ For 'g(f(-1))', substitute '3x - 5' for 'x' in the function 'g', resulting in '(3x - 5)^2 + 2'.
- 𧟠Simplify 'g(f(-1))' to '(3*(-1) - 5)^2 + 2', which equals '(-3 - 5)^2 + 2' or '(-8)^2 + 2', resulting in 64 + 2 = 66.
- đ The video emphasizes that the order of functions in composite notation determines which function is the input.
- đŻ The video aims to teach viewers how to evaluate composite functions and encourages them to like, subscribe, and hit the bell for updates.
Q & A
What is a composite function?
-A composite function is a function that is formed by applying one function to the result of another function. It is denoted as f(g(x)) or f circle of g of x.
How is f(g(x)) different from g(f(x))?
-f(g(x)) means that the function g(x) is used as the input for the function f, while g(f(x)) means that the function f(x) is used as the input for the function g. The order of the functions matters.
What are the given functions in the transcript?
-The given functions are f(x) = 3x - 5 and g(x) = x^2 + 2.
How is f(g(x)) evaluated when x is replaced by 2?
-To evaluate f(g(2)), you first calculate g(2) which is 2^2 + 2 = 6, then substitute this into f(x) to get f(6) = 3*6 - 5 = 18 - 5 = 13.
What is the result of f(g(2))?
-The result of f(g(2)) is 13.
How is g(f(x)) evaluated when x is replaced by -1?
-To evaluate g(f(-1)), you first calculate f(-1) which is 3*(-1) - 5 = -8, then substitute this into g(x) to get g(-8) = (-8)^2 + 2 = 64 + 2 = 66.
What is the result of g(f(-1))?
-The result of g(f(-1)) is 66.
What does the notation f circle of g of x mean?
-The notation f circle of g of x means the composition of function g with function f, which is read as 'f composite g of x'.
Why is it important to follow the order of functions when evaluating composite functions?
-The order of functions is important because it determines which function's output becomes the input for the other function. Changing the order can lead to different results.
What is the significance of the variable x being replaced by another function in composite functions?
-In composite functions, replacing the variable x with another function allows for the application of one function's output to another function, which is the core concept of function composition.
How can one avoid confusion between the notations f(g(x)) and g(f(x))?
-To avoid confusion, remember that the function that comes first in the notation is the outer function, and the one that comes second is the inner function whose result is used as the input for the outer function.
Outlines
đ Introduction to Composite Functions
The speaker, Teacher, introduces the concept of composite functions, which are functions within functions. The notation f(g(x)) or f â g(x) is explained, where 'f' is the outer function and 'g' is the inner function. The speaker emphasizes that the order of functions indicates the order of operation, with 'f' being applied after 'g'. The video then presents a problem involving two functions: f(x) = 3x - 5 and g(x) = x^2 + 2. The task is to evaluate f(g(2)) and g(f(-1)). The speaker begins by evaluating f(g(x)) by substituting g(x) into f(x), resulting in 3(x^2 + 2) - 5. The substitution is then made with x = 2, leading to 3(2^2 + 2) - 5, which simplifies to 3(4 + 2) - 5 = 3(6) - 5 = 18 - 5 = 13. The speaker clarifies that in composite functions, the function that appears first in the notation is applied last to the input.
đ Evaluating Composite Functions: Part 2
Continuing from the previous explanation, the speaker now evaluates g(f(-1)). This involves substituting f(x) into g(x), resulting in (3x - 5)^2 + 2. The substitution is made with x = -1, leading to (3(-1) - 5)^2 + 2, which simplifies to (-3 - 5)^2 + 2 = (-8)^2 + 2 = 64 + 2 = 66. The speaker concludes by summarizing the results: f(g(2)) = 13 and g(f(-1)) = 66. The video aims to educate viewers on how to evaluate composite functions and encourages new subscribers to like, subscribe, and enable notifications for updates. The speaker signs off with their name, Turgon.
Mindmap
Keywords
đĄcomposite function
đĄfunction composition
đĄnotation
đĄinput function
đĄevaluation
đĄvariable substitution
đĄfunction f
đĄfunction g
đĄsimplification
đĄoperation order
đĄexample problems
Highlights
Introduction to composite functions
Notation for composite functions explained
Understanding the order of functions in composite notation
Given functions f(x) = 3x - 5 and g(x) = x^2 + 2
Evaluating f(g(2)) step by step
Substituting g(x) into f(x) for f(g(2))
Simplifying the expression for f(g(2))
Final calculation for f(g(2)) equals 13
Clarification on the difference between f(g(x)) and g(f(x))
Evaluating g(f(-1)) step by step
Substituting f(x) into g(x) for g(f(-1))
Simplifying the expression for g(f(-1))
Final calculation for g(f(-1)) equals 66
Emphasis on the importance of function order in composite functions
Encouragement for new viewers to subscribe and engage with the channel
Transcripts
hi guys it's me teacher going in today's
video we will talk about the composite
function or composition of function
so we have here this notation
this is read as f
circle of G of X or
F composite G of X and that is equal to
this notation f g of X we're in in this
function since the variable F or the
function f comes first before G
G will be the input or the input
function
with respect to the function f so
without further Ado let's do this topic
so let's have this problem
we are given two different functions f
of x is equal to 3x minus 5 and the
other is 3x at 3 of x g of x
is equal to x squared plus two and right
now we are asked to
evaluate this one we have f
of G of 2 and the other is G of f of
negative one let's start with number one
in this case RF of G of 2 here
first we will try to evaluate
f
of G
of X that is the same as that
okay now what will happen here is that
in this case guys
we will move this one
since
your function f
comes first before G
we will plug in
this x square plus 2
to replace the variable X in
3x minus five so it goes like this three
copy three
then your X this x will be replaced by x
square
times x square
plus 2
minus 5. so here what is the reason Kai
that is f
look at this one your gfx is inside the
parenthesis this will be the input
so we will replace the variable x 3
times x square
plus 2 minus 5. so what will happen
after that is as you can see
is that we will replace this variable X
by 2 and it will become
3
times
2 square
plus 2 minus 5. so what will happen is
that you need to simplify this to become
3
times 2 squares four
plus two then we have minus five
simplify
three
times four plus two which is equal to
six
and then minus 5.
so this is three times six is eighteen
minus five therefore
your f
of G
of 2 is equal to 15 minus 18 minus 5
which is equal to 13. guys don't be
confused about these two different
notations if the variable F comes first
meaning your input is
the function G
if the function G comes first
your function f will be the input okay
let's continue
here number two we have G of f of
negative one
so it will become
G
off
f
of X it goes like this meaning
we will use this and this 3x minus 5
will be replaced or we will be using
this to substitute for the value
variable X and it will become like this
x squared that is x squared by copy this
one
3x minus 5
Square
and then don't forget plus two sure
X squaring
now after doing that remember that you
are asked to evaluate negative one
we will replace this x by negative one
it will become
3
times negative one
minus 5 square plus two
so simplify that into three times
negative 1 is negative
three
minus five
then square plus 2. so simplify that
until your negative three minus five is
negative eight
Square
plus 2 and negative 8
squared is not negative 64. right the
answer is 60 4 and then plus 2.
therefore
your G
can you say this here G
of f
of negative 1
is equal to 64 plus 1 and that is 66.
and this is the answer guys
so I hope guys learned something from
this video on how to
evaluate composite functions so if
you're new to my channel don't forget to
like And subscribe button hit the Bell
button for you to be updated starting
latest uploads again it's me to turgon
my name is
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