Compositional Functions: Formulas, Properties and Example Problems
Compositional Functions: Formulas, Properties and Example Problems – What is meant by composition function? At this time About the knowledge.co.id will discuss about the Composition Function and the things that surround it. Let's look at the discussion together in the article below to better understand it.
Compositional Functions: Formulas, Properties and Example Problems
The composition function is a combination of operations on two types of functions f (x) and g (x) to produce a new function. The usual compositional function operations are denoted by an “o” and read by composition or roundabout.
The new functions that can be formed from f (x) and g (x) are:
(f o g)(x) = g put in f
(g o f)(x) = f plug in to g
That single function is a function that can be denoted by the letters "f o g" or can also be read by "function f roundabout g". The function "f o g" is the function g which is done first and then followed by f. Meanwhile, the function "g of f" is read with the roundabout g function f. So, "g o f" is a function where f is done before g.
Composition Function Formulas
From this formula, the definition obtained is:
If f: A → B is determined by the formula y = f (x)
If g: B → C is determined by the formula y = g (x)
So, the results of the functions g and f :
h (x) = (g o f)(x) = g( f (x))
From this explanation it can be concluded that functions involving functions f and g can be written:
(g o f)(x) = g (f (x))
(f o g)(x) = f (g (x))
Properties Functions Composition
In the following we will provide some of the properties of the composition function, including the following:
If f: A → B, g: B → C, h: C → D, several properties such as:
(f o g)(x)≠(g o f)(x). The commutative property does not apply.
[f o (g o h)(x)] = [(f o g ) o h (x)]. Will be associative.
If the identity function is I(x), then (f o l)(x) = (l o f)(x) = f (x).
The Function of Composition in Life
The definition of the function of composition in life has examples of the function of composition found in everyday life, the explanation is as follows:
- Through the process of producing a book can be done in two steps, namely:
- The first step is editing and then it will go directly to the production step.
- At the time of the editing step, the manuscript will be immediately edited and in the layout to change as a file that is ready to be printed.
- Furthermore, the file can be immediately processed to the printing production step so that it can be 100% into a book.
- The process of producing this book is using the application of the compositional function algorithm.
- As a metal recycling process, namely:
- For the initial process, several mixed metal shards will be put together so that they can become small flakes.
- Furthermore, the magnetic drum inside the crusher can distinguish magnetic metal which has an element of iron from other elements.
- Next, the remaining metal fragments are dredged and separated. And for iron flakes can be immediately melted into new steel. This metal recycling process is using the application of the composition function.
Examples of Composition Function Questions
- Problem 1
Given two functions, respectively f (x) and g (x) respectively, namely:
f(x) = 3x + 2
g(x) = 2 − x
Define:
a) (f o g) (x)
b) (g o f) (x)
Answer
Data:
f(x) = 3x + 2
g(x) = 2 − x
a) (f o g)(x)
“Enter the g (x) kef (x)”
until it becomes:
(f o g)(x) = f ( g (x) )
= f (2 − x)
= 3 (2 − x) + 2
= 6 − 3x + 2
= − 3x + 8
b) (g o f ) (x)
“Put the f (x) into g (x)”
Until it becomes:
(f o g) (x) = g (f (x) )
= g( 3x + 2)
= 2 − ( 3x + 2)
= 2 − 3x − 2
= − 3x
- Problem 2
If you know that f (x) = 3x + 4 and g (x) = 3x, what is the value of (f o g) (2)?
Answer:
(f o g) (x) = f (g (x))
= 3(3x) + 4
= 9x + 4
(f o g) (2) = 9(2) + 4
= 22
- Problem 3
It is known that the function f (x) = 3x − 1 and g (x) = 2×2 + 3. The value of the function composition ( g o f )(1) =….?
A 12
B. 8
C. 7
D. 11
E. 9
Answer
Is known:
f (x) = 3x − 1 and g (x) = 2×2 + 3
( g o f )(1) =…?
Enter the f (x) in g (x) then fill it with 1
(g o f) (x) = 2 (3 x − 1) 2 + 3
(g o f) (x) = 2 (9 x 2 − 6x + 1) + 3
(g o f) (x) = 18x 2 − 12x + 2 + 3
(g o f) (x) = 18×2 − 12x + 5
(g o f) (1) = 18 (1) 2 − 12(1) + 5 = 11
- Problem 4
Given two functions:
f(x) = 2x − 3
g(x) = x2 + 2x + 3
If (f o g)(a) is 33, determine the value of 5a
Answer:
Find first (f o g)(x)
(f o g)(x) equals 2(x2 + 2x + 3) − 3
(f o g)(x) equals 2×2 4x + 6 − 3
(f o g)(x) equals 2×2 4x + 3
33 equals 2a2 4a + 3
2a2 4a − 30 equals 0
a2 + 2a − 15 equals 0
Factor:
(a + 5)(a − 3) equals 0
a = − 5 or a equal to 3
Until
5a = 5(−5) = −25 or 5a = 5(3) = 15
- Problem 5
If (f o g)(x) = x² + 3x + 4 and g (x) = 4x – 5. What is the value of f (3)?
Answer:
(f o g)(x) equals x² + 3x + 4
f(g(x)) equals x² + 3x + 4
g(x) equals 3 So,
4x – 5 equals 3
4x equals 8
x equals 2
f (g (x)) = x² + 3x + 4 and for g (x) equals 3 we get x equals 2
Up to: f (3) = 2² + 3. 2 + 4 = 4 + 6 + 4 = 14
It is known that the function f (x) = 3x − 1 and g (x) = 2x2 + 3. The value of the function composition (g o f)(1) =….
A 7
B. 9
C. 11
D. 14
E. 17
Answer:
Is known:
- f (x) = 3x − 1 and g (x) = 2x2 + 3
Asked:
- (g o f)(1) =…….
Enter the f (x) in g (x) then fill it with the number 1, so it will become:
(g o f)(x) = 2(3x − 1)2 + 3
(g o f)(x) = 2(9x2 − 6x + 1) + 3
(g o f)(x) = 18x2 − 12x + 2 + 3
(g o f)(x) = 18x2 − 12x + 5
(g o f)(1) = 18(1)2 − 12(1) + 5 = 11
Answer: C
- Problem 6
Known one f -1 (4x-5) = 3x-1 and (f-1 ◦ f)(5)= p2 +2p – 10 then the average of the p values is…
a. -4
b. -2
c. -1
d. 1
e. 4
Answer:
f (x) = y ↔ f-1 (y) = x
f (5) = y
f–1 (4x-5) = 3x-1
So we will get 3x-1 = 5
x = 2 and y = 4x-5 = 3
x = 2
Determine the p-value
(f– -1 ◦ f)(5) = p2 +2p-10
f -1 (f(5)) = p2 +2p – 10
f—1(3) = p2 +2p – 10
3(2)-1 = p2 +2p – 10
p.s2 + 2p – 1 = 0
(p + 5)(p – 3) = 0
p = -5 and p = 3
So, the average p-value is (-5) + 3 / 2 = -1
Answer: C
Thus the review from About the knowledge.co.id about Composition Function , hopefully can add to your insight and knowledge. Thank you for visiting and don't forget to read other articles
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