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High School Mathematics - 2
4.6 Composition of Functions

The basic idea of functions is that kind of a chain reaction in which the functions occur one after the other.
Let f be a function whic associates with each person that person's mother and g be a function which associates with each person that person's father. Then
f = {(x,y): x is a person and y is the mother of x}
and g = {(x,y): x is a person and y is the father of x}
We compose f with g by "applying" f and then g.

We shall use the symbol "g(f)" to denote the composition of f with g.
f(John) = John's mother Mary and g (Mary) = Mary's mother
Thus g(Mary) = g(f(John)) and in the sense the "g" comes before "f". Therefore, we say that g(f)(John) = g(f(John)) = John's maternal grandfather.
Similarly, f(g(Hari)) = f(Peter) = Peter's mother = John's paternal grandmother.

Definition: Given two functions, f and g, the function x -> g(f(x)) is called a composite of f and g and is denoted by gf. The domain of gf is the set of all elements x in the domain of f for which f(x) is in the domain of g. The operation of formng a composite of two functions is called composition.
The function g(f(x)) is also denoted by (gof)(x) and is read as 'g circle f'. Notice that the order of the symbolsis the same as in g(f(x), i.e, f is applied first, then g.
The composition of functions is associative.

Example: Given that f: x -> 3x-2 and g:x ->x5 for all x R, find (gf)(x)
(gf)(x) = g(f(x) = g(3x-2) = (3x-2)5
Solution: (3x-2)5


Directions: Find the composite functions of the following.
Q 1: If f: x -> 2x, g: x -> x2 and h: x -> x +1. Find the formula for ho(gof)
4x+1
4x
4x2+1

Q 2: Given that f: x -> 3x-2 and g:x ->x5 for all x R, find f(g(x) -f(x))
(3x-2)5
3x5-9x+4
(2x-3)5

Q 3: If f: R -> R, g: R -> R are functions defined by f(x) = 3x -2 , g(x) - x2 + 1, find (gof-1)(2)
25/9
1/2
4/3

Q 4: Is g(x) x f(x) same as g(f(x)?
yes
No

Q 5: Given that f: x -> 3x-2 and g:x ->x5 for all x R, find (ff)(x).
3x-2
(3x-2)5
9x-8

Q 6: Given f(x) = 3x - 2, find f(x-3).
3x - 11
3x - 7
3x-5

Q 7: Given that f: x -> 3x-2 and g:x ->x5 for all x R, find f(g(x)+3)
3x5+7
(3x-2)5
9x-8

Q 8: If z = g(y) = 3y2 - 2y + 1 and y = f(x) = 4x+7, find the composite function gof.
(4x+5)2
4x+7
48x2 + 160x + 134

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Question 10: This question is available to subscribers only!


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