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Composition of Functions | Mathematics (Maths) Class 12 - JEE PDF Download

Composite Functions

Let f:  X → Y1 and g: Y→ Z be two functions and the set D = {x ∈ X: f(x) ∈ Y2}. If D ≡ Ø, then the function h defined on D by h(x) = g{f(x)} is called composite function of g and f and is denoted by gof. It is also called function of a function.

Composition of Functions | Mathematics (Maths) Class 12 - JEE

Remark: Domain of gof is D which is a subset of X (the domain of f). Range of gof is a subset of the range of g. If D = X, then f(X) ⊆ Y2.

Properties of composite functions

  1. The composite of functions is not commutative i.e. gof ≠ fog.
  2. The composite of functions is associative i.e. if f, g, h are three functions such that fo (goh) & (fog) oh are defined, then fo (goh) = (fog) oh.

Example 1. Let f(x) = ex ; R+ → R and g(x) = sin-1 x; [-1, 1] →[- π/2 ,  π/2]. Find domain and range of fog (x)
Solution. Domain of f(x): (0, ∞), Range of g(x): [- π/2 ,  π/2]
The values in range of g(x) which are accepted by f(x) are (0, π/2]
⇒ 0 < g(x) ≤ π/2   0 < sin-1 x ≤ π/2 , 0 < x ≤ 1
Hence domain of fog(x) is x ∈ (0, 1]

Composition of Functions | Mathematics (Maths) Class 12 - JEE

Example 2. Let f(x) = (x-1)/(x+1) ,  f2(x) = f{f(x)}, f3 (x) = f{f2(x)},.....fk + 1 (x) = f{fk(x)}. for k = 1, 2, 3,...., Find f1998 (x).
Solution.
Composition of Functions | Mathematics (Maths) Class 12 - JEE

Composition of Functions | Mathematics (Maths) Class 12 - JEE

Composition of Functions | Mathematics (Maths) Class 12 - JEE

Composition of Functions | Mathematics (Maths) Class 12 - JEE
Thus, we can see that fk(x) repeats itself at intervals of k = 4.
Hence, we have f1998(x) = f2(x) = -1/x, [∴ 1998 = 499 × 4 + 2]

Example 3. Let g : R → R be given by g(x) = 3 + 4x. If gn(x) = gogo....og(x), show that fn(x) = (4n - 1) + 4nx if g-n (x) denotes the inverse of gn (x).
Solution. Since g(x) = 3 + 4x
g2(x) = (gog) (x) = g {g (x)} = g (3 + 4x) = 3 + 4 (3 + 4x) or g2(x) = 15 + 42x = (42 – 1) + 42x
Now g3(x) = (gogog) x = g {g2 (x)} = g (15 + 42 x) = 3 + 4 (15 + 42 x) = 63 + 43 x = (43 –1) + 43x
Similarly we get gn(x) = (4n – 1) + 4nx
Now let g(x) = y  ⇒ x = g–n(y) ....(1)
∴ y = (4n – 1) + 4nx or x = (y + 1 – 4n)4–n ...(2)
From (1) and (2) we get g–n (y) = (y + 1 – 4n) 4–n.
Hence g–n (x) = (x + 1 – 4n) 4–n

Example 4. If f(x) = If f(x)  = | |x – 3| – 2 | ; 0 ≤ x ≤  4 and g(x) = 4 – |2 – x| ; –1 ≤  x ≤  3 then find fog(x).
Solution.
  Composition of Functions | Mathematics (Maths) Class 12 - JEE
Composition of Functions | Mathematics (Maths) Class 12 - JEE
Composition of Functions | Mathematics (Maths) Class 12 - JEE

Composition of Functions | Mathematics (Maths) Class 12 - JEE
Composition of Functions | Mathematics (Maths) Class 12 - JEE
Composition of Functions | Mathematics (Maths) Class 12 - JEE

Example 5. Prove that f(n) = 1 - n is the only integer valued function defined on integers such that
(i) f(f(n)) = n for all n ε Z and 
(ii) f(f(n + 2) + 2) = n for all n ε Z and 
(iii) f(0) = 1.
Solution. The function f(n) = 1 - n clearly satisfies conditions (i), (ii) and (iii). Conversely, suppose a function
f: Z → Z satisfies (i), (ii) and (iii). Applying f to (ii) we get, f(f(f(n + 2) + 2))) = f(n) and this gives because of (i), f(n + 2) + 2 = f(n), ........(1)
for all n ε Z. Now using (1) it is easy to prove by induction on n that for all n ε Z,
Composition of Functions | Mathematics (Maths) Class 12 - JEE
Also by (iii), f(0) = 1. Hence by (i), f(1) = 0. Hence f(n) = 1 - n for all n ε Z.

Question for Composition of Functions
Try yourself:
Which of the following statements about composite functions is true?
View Solution

General Definition

  • Identity function: A function f ; A → defined by f(x) = x ∀ x ∈ A is called the identity of A & denoted by IA.
    Ex: f : R+ → R+ ; f(x) = eℓnx and f : R → R ; f(x) = ℓn ex
    Every Identity function is a bijection.
  • Constant function: A function f: A → B is said to be constant function. If every element of set A has the same functional image in set B i.e. f: A → B ; f(x) = c ∀ x ε A & c ε B is called constant function.
  • Homogeneous function: A function is said to be homogeneous w.r.t. any set of variables when each of its term is of the same degree w.r.t. those variables.
  • Bounded Function: A function y = f(x) is said to be bounded if it can be express is the form of
    a ≤ f(x) ≤ b where a and b are finite quantities.
    Example: -1 ≤ sin x ≤ 1 ; 0 ≤ {x} < 1 ; -1 ≤ sgn (x) ≤ 1 but ex is not bounded.
    Any function having singleton range like constant function.
  • Implicit function & Explicit function: If y has been expressed entirely in terms of `x' then it is called an explicit function.
    If x & y are written together in the form of an equation then it is known as implicit equation corresponding to each implicit equation there can be one, two or more explicit function satisfying it
    Example: 
    y = x3 + 4x2 + 5x → Explicit function
    x + y = 1 → Implicit equation
    y = 1 - x → Explicit function
  • Even & Odd Functions
    Function must be defined in symmetric interval [-x, x]
    If f (-x) = f (x) for all x in the domain of `f' then f is said to be an even function.
    e.g. f (x) = cos x; g (x) = x² + 3.
    If f (-x) = -f (x) for all x in the domain of `f' then f is said to be an odd function.
    e.g. f (x) = sin x ; g (x) = x+ x.

Remark

  • f (x) - f (-x) = 0 ⇒ f (x) is even & f (x) + f (-x) = 0 ⇒ f (x) is odd.
  • A function may be neither even nor odd.
  • Inverse of an even function is not defined.
  • Every even function is symmetric about the y-axis & every odd function is symmetric about the origin.
  • A function (whose domain is symmetric about origin) can be expressed as a sum of an even & an odd function. e.g.
    Composition of Functions | Mathematics (Maths) Class 12 - JEE 
  • The only function which is defined on the entire number line & is even and odd at the same time is f(x) = 0.
  • If f and g both are even or both are odd then the function f.g will be even but if any one of them is odd and other even then f.g will be odd.

Example 6. Which of the following functions is odd ?
(a) sgn x + x2000  
(b) | x | - tan x 
(c) x3 cot x 
(d) cosec x55
Solution.
Let’s name the function of the parts (A), (B), (C) and (D) as f(x), g(x), h(x) & f(x) respectively. Now
(a) f(–x) = sgn (–x) + (–x)2000 = –sgn x + x2000 ≠ f(x) & ≠ –f(x) ∴  f is neither even nor odd.
(b) g(–x) = |–x| – tan (–x) = |x| + tan x ∴ g is neither even nor odd.
(c) h(–x) = (–x)3 cot (–x) = –x3 (–cot x) = x3 cot x = h(x) ∴ h is an even function
(d) f(–x) = cosec (–x)55 = cosec (–x55) = –cosec x55 = – f(x) ∴ f is an odd function.
Alternatively
(a) f(x) = sgn (x) + x2000 = O + E = neither E nor O
(b) g(x) = E – O = Neither E nor O
(c) h(x) = O × O = E (D) f(–x) = O o O = O
(d) is the correct option

Example 7. 
Composition of Functions | Mathematics (Maths) Class 12 - JEE
(a) An even function 
(b) An odd function
(c) Neither even nor odd function 
(d) None of these
Solution.
Composition of Functions | Mathematics (Maths) Class 12 - JEE
= O × eO × O = O × eE = O × E = O

Example 8. Let f: [-2, 2] → R be a function if f(x) = 
Composition of Functions | Mathematics (Maths) Class 12 - JEE
 so that
(i) f is an odd function 
(ii) f is an even function (where [*] denotes the greatest integer function)
Solution.
Composition of Functions | Mathematics (Maths) Class 12 - JEE
Composition of Functions | Mathematics (Maths) Class 12 - JEE
Composition of Functions | Mathematics (Maths) Class 12 - JEE
(i) If f is an odd function then f(x) = –f (–x)
Composition of Functions | Mathematics (Maths) Class 12 - JEE
(ii) If f is an even function  
Composition of Functions | Mathematics (Maths) Class 12 - JEE

Example 9. Let f(x) = ex + sin x be defined on the interval [-4, 0]. Find the odd and even extension of f(x) in the interval [-4, 4].
Solution.
Odd Extension:  Let g0 be the odd extension of f(x), then
Composition of Functions | Mathematics (Maths) Class 12 - JEE
Even Extension:  Let ge be the odd extension of f(x), then
Composition of Functions | Mathematics (Maths) Class 12 - JEE

The document Composition of Functions | Mathematics (Maths) Class 12 - JEE is a part of the JEE Course Mathematics (Maths) Class 12.
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FAQs on Composition of Functions - Mathematics (Maths) Class 12 - JEE

1. What is a composite function?
Ans. A composite function is the combination of two functions where the output of one function becomes the input of another function.
2. How do you find the composite of two functions?
Ans. To find the composite of two functions f(x) and g(x), you can simply plug g(x) into f(x) or vice versa, depending on the order specified.
3. What is the notation for composite functions?
Ans. The notation for composite functions is (f ∘ g)(x), where f and g are the functions being composed.
4. How can composite functions be used in real-life applications?
Ans. Composite functions are commonly used in fields like economics and physics to model complex relationships between different variables.
5. Can you have a composite function of more than two functions?
Ans. Yes, you can have a composite function of more than two functions by repeatedly composing two functions at a time.
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