Composition of Function by Ma'am Ella Barrun
Summary
TLDRThis video script introduces the concept of function composition, explaining it as substituting one function into another to create a new function. It demonstrates this with several examples, showing step-by-step calculations for composite functions such as F(G(x)) and G(F(x)). The script covers linear and quadratic functions, illustrating the process of finding composite functions and their simplified forms, including evaluating composite functions at specific points.
Takeaways
- 📚 The concept of function composition is introduced, where one function is substituted into another to create a new function.
- 🔢 The notation for function composition is explained, using f(g(x)) to represent the composition of function f with function g.
- 📝 An example is provided to illustrate the process of function composition, starting with simple linear functions and moving to more complex polynomials.
- 📉 The first example demonstrates the composition of two linear functions, resulting in a new linear function.
- 📚 The second example involves composing a linear function with a quadratic function, resulting in a new quadratic function.
- 🔑 The third example shows the composition of a quadratic function with another quadratic function, leading to a new quadratic expression.
- 📌 The fourth example introduces the concept of evaluating a composite function at a specific value, in this case, G(-4).
- 🧩 The process of substituting the result of one function into another is shown step by step, with simplification of the resulting expression.
- 🔍 An example of finding the square root of a composite function's result is given, highlighting the application of real-world operations on function compositions.
- 📈 The final example demonstrates the composition of a quadratic function with a linear function, resulting in a new quadratic expression when evaluated at a specific point.
- 🤔 The script concludes with an invitation for questions, indicating an interactive component to the lesson on function composition.
Q & A
What is the definition of function composition?
-Function composition is the process of substituting one function into another, creating a new function. If a function 'f' is substituted into all variables of another function 'g', it is denoted as 'f composite with g' or 'f(g(x))'.
How is the composition of functions denoted mathematically?
-The composition of functions is denoted as 'f ∘ g' or 'f(g(x))', where 'f' and 'g' are functions and 'x' is the variable.
What is the first example of function composition given in the script?
-The first example given is G(x) = 5x + 3 and F(x) = 2x + 1. The composition of F with G, denoted as F(G(x)), is calculated as 2*(5x + 3) + 1, which simplifies to 10x + 7.
In the second example, what are the functions F(x) and G(x), and what is their composition?
-In the second example, G(x) is given as 5x^2 - x + 4 and F(x) as 2x - 1. The composition F(G(x)) simplifies to 10x^2 - 2x + 7.
What is the third example of function composition in the script?
-In the third example, F(x) = 2x - 15, G(x) = x^2 + 19x + 90. The composition G(F(x)) simplifies to 4x^2 - 22x + 30.
What is the purpose of the fourth example in the script?
-The fourth example demonstrates how to find the composition of the function H with G at a specific value, x = -4. It shows the process of substituting the result of G(-4) into H(x) and finding the final value.
What is the result of the composition H(G(-4)) in the fourth example?
-The result of the composition H(G(-4)) is 6, after simplifying the expressions and evaluating the square root of the final value.
What is the function F(x) in the last example of the script?
-In the last example, the function F(x) is given as x^2 + 5x + 6.
How is the composition F(G(4)) calculated in the last example?
-In the last example, after finding G(4) which equals 6, this value is substituted into F(x), resulting in F(6) which simplifies to 72.
What is the main takeaway from the script regarding function composition?
-The main takeaway is understanding how to perform function composition by substituting one function into another, and simplifying the resulting expression to obtain a new function or a specific value.
Outlines
📚 Introduction to Function Composition
The script begins by transitioning from the topic of operations on functions to the concept of function composition. It defines function composition as the process of substituting one function into another, resulting in a new function. The notation for this is f composed with g, denoted as f(g(x)). The first example illustrates this with f(x) = 2x + 1 and g(x) = 5x + 3, showing the step-by-step process of finding f(g(x)) which simplifies to 10x + 7.
🔍 Examples of Function Composition
This paragraph provides further examples to demonstrate function composition. It includes the process of finding f(g(x)) and g(f(x)) for different functions, such as f(x) = 2x - 1 and g(x) = x^2 - x + 4, and another with f(x) = 2x - 15 and g(x) = x^2 + 19x + 90. Each example is worked through, applying the distributive law and simplifying the results to show the composed functions.
📘 Applying Composition with Specific Values
The final paragraph explores the application of function composition with specific values, starting with an example where f(x) = x^2 + 5x + 6 and g(x) = x + 2. It calculates the composition of f with g evaluated at x = 4, resulting in f(g(4)) = 72. The script also includes an example involving the square root function h(x) = sqrt(x + 6), where h is composed with g at x = -4, yielding h(g(-4)) = 6.
Mindmap
Keywords
💡Function Composition
💡Variable
💡Distributive Law
💡Simplification
💡Algebraic Expression
💡Quadratic Function
💡Composite Function
💡Square Root
💡Polynomial
💡Binomial
Highlights
Definition of function composition: substituting one function into all variables of another function to create a new function.
Notation for function composition: f composite with g, denoted as f(g(x)).
Example 1: Composition of linear functions g(x) = 5x + 3 and f(x) = 2x + 1 results in f(g(x)) = 10x + 7.
Example 2: Composition of quadratic and linear functions, resulting in a quadratic function f(g(x)) = 10x^2 - 2x + 7.
Example 3: Composition of linear functions to get a quadratic function g(f(x)) = 4x^2 - 22x + 30.
Example 4: Composition involving square root function h(g(x)) = sqrt(x + 6), demonstrating real-world applications.
Finding h(g(-4)) = 6 by substituting -4 into the composed function.
Example 5: Composition of quadratic and linear functions to find f(g(4)) = 72.
Step-by-step process of function composition illustrated through multiple examples.
Use of distributive law in simplifying the composed functions.
Importance of function composition in creating new functions from existing ones.
Practical demonstration of how to perform function composition with clear examples.
Clarification of the difference between f(g(x)) and g(f(x)) in function composition.
Emphasis on the algebraic manipulation required for simplifying composed functions.
Explanation of how to handle different types of functions in composition, including linear, quadratic, and square root functions.
Highlighting the final simplified form of each composed function as the key takeaway.
Invitation for questions or messages at the end of the discussion, promoting engagement.
Transcripts
okay so since we are done with operation
of function let us now proceed to
composition of function but first let us
Define what is composition of function
okay so if a function is substituted to
all variables in another function then
you are performing a composition of
function to create another function
Okay so
for compition
function f composite with G or equals
okay
equivalent F of G of X so G of X is
insert f of x to create another function
okay so let us now have an example
number
one so we have here uh G of X is = 5x +
3 and then F ofx down is 2x +
1 okay let's just
copy F composite with G of X or
equivalent or equal F of G of x
okay so let us find F composite with
G
okay okay
soag
is um
identify g of x g of X is 5x + 3
okay okay so mag g f time
5x +
3
so FX that is 2x +
1 G of
xert or
inut FX
so 5x +
3it xay mag 2 X or sorry 2 * 5x + 3 + 1
okay so
p x 5x + 3 Yun G of X and then let us
simplify 2 * 5x that that is 10 x and
then 2 * 3 that is 6 and then let's just
cop
then and then 6 + 1 that is POS 7
so F composite with G is 10 x + 7 okay
so now let us proceed to example number
two so example number
G of x = 5 x 2 - x + 4 and then F ofx
that is 2x
-1 okay F composite with G of X or
equivalent F of G of X
so F composite with G of
X Okay so soay an value G of X that is
5X
2us x + 4 and
then
FX okay
2x-1 so
X
mam 5x 2 - x + 4 okay
so okay mag 2 * 5X x^ 2 - x + 4 - 1 and
then s is
simplify okay 2 * 5 x 2 that is 10
X2 and then 2
*
X is -2X and then 2 * pos4 that is
postive 8 and then let's just copy
1 and then
simplify okay so 10
x² and then um let's just copy
alsox and then last
8us POS 7
so um f composite with G of x = 10 x^ 2
- 2x + 7 okay so to example number three
f of x is = 2x -
15 p g of X = X2 + 19 x +
90 and
then let us find G composite with f of
x
so FX and then
insert G of
X
okay
so
insert f of x
2x - 15 so s g of x
2x - 15 so maging 2x - 15
2 + 19 * 2x -
15 + 90 okay and then let
us um
simplify okay so for the first binomial
let let us use distributive
law
4x2 -
60x +
225 and then next 19 * 2x that is
positive
38x and then next 19
* 15 that will
be 285 and then you uh positive 9
copy
Nang okay so let us simplify
again okay so
since 4X let us just
copy and then NE
60x is POS 38x
so
22x and then
lastly um constant number that is or
positive
225 minus
285 + 90 so that will be postive 30 so G
of x uh sorry G of f ofx or you g
composite with f ofx is = to 4 x^2 -
22x + 30
okay
so example number
four okay so example
or given FX = 2x^ 2 - 15 and then G of X
is x² + 19 x + 90 and then h of X is squ
root x +
6 and then let us find G uh let us find
H composite with G
of4
number so
unin G of-4 Kasi po Yan
equivalent h
h of
G of
negative okay
so G of X
is
x² + 19 x + 90
soitan -4 since
X
is
X
so -4 -4 2 + 19 * -4 +
90 so let us
simplify
so
um -4 * -4 that is POS POS 16
and
then um positive 19 * -4 that will
be76 and then just copy positive 90 and
then
simplify 16 - 76 + 90 so that will be
postive 30 okay so proceed Naman
H
of4
since G
of4 okay so h of X is s otk x +
30 so
since G
of4 which is
30
insert X
H
of4 okay
soqu < TK of 30 +
6 okay and then 30 + 6 that is 36 or
square root of 36 so square root of 36
is
6 h composite of G or composite with G
of -4 is equal to 6
okay so let us proceed to our last
example so we have here f of x = X2 + 5x
+
6 okay and then G of x = x +
2 so let let us
find F composite with G of 4
or F of G of
4 so example number four that
is4 X which is
pos4 okay so
unin G of
4 okay so G of X is x + 2 so
x
a
given so magig 4
+
2 so mag G of 4 is equals to 6 since
G of 4 which is
6 insert or
input f of x f of x is x² + 5x + 6
so
is variable or X okay so mag
F ofx = 6 2 + 5 * 6 + 6 and then let us
simplify 6 * 6 that is 36 and then plus
5 * 6 that is 30 and then we just need
to copy 6 and then simplify again 36 +
30 that is 6 6 + 6 that is
72 excuse me
so F composite with G of 4 is equals to
72
okay so a
lang if there's any question
um um a message okay
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