•• CLASSIFICATION OF FUNCTIONS••
*****************""""
A) ONE ONE FUNCTION (INJECTIVE MAPPING):
A function f: A--> B is said to be a one-one function or injective mapping if different elements of A have different f images in B. Thus for x₁, x₂∈ A & f(x₁), f(x₂)∈ B. f(x₁)= f(x₂) <=> x₁ = x₂ or x₁≠ x₂ <=> f(x₁) ≠ f(x₂).
****NOTES
1) Any continuous function which is entirely increasing or decreasing in whole domain is one-one.
2) If a function is one-one, any line parallel to x-axis cuts the graph of the function at most one point.
ALGORITHM
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STEP-1: Take two arbitrary elements x,y(say) in the domain of f.
STEP-2: Put f(x)= f(y)
STEP-3: Solve f(x)= f(y)
If f(x)= f(y) gives x= y only, then f: A--> B is a one-one function (or injection) otherwise not
NOTE:
Let f: A--> B and let x,y belongs to A. Then, x = y => f(x)= f(y) is always true from the definition, But f(x)= f(y) => x= y is true only when f is one-one.
MANY-ONE FUNCTION:
A function f: A--> B is said to be a many one function if two or more elements of A have the same f image in B.
Thus f: A-->B is many one if Ξ| x₁, x₂∈ A, f(x₁)= f(x₂) but x₁ ≠ x₂
**NOTE
If a continuous function has local maximum or local minimum, then f(x) is many-one because atleast one line parallel to x axis will intersect the graph of function at least twice.
ALGORITHM
-----------------
STEP-1: Take two arbitrary elements x,y(say) in the domain of f.
STEP-2: Put f(x)= f(y) and solve the equation.
STEP-3: Solve f(x)= f(y)
If f(x)≠ f(y) gives x≠ y only, then f: A--> B is a many-one function otherwise not.
NOTE:
Recall that if we get x= y, then f is a one-one function
ONTO FUNCTION(SURJECTION) :
if range = co-domain, then f(x) is onto.
ALGORITHM
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STEP-1: Choose an arbitrary element y in B.
STEP-2: Put f(x)= y
STEP-3: Solve f(x)= y for x and obtain x terms of y. Let x = g(y)
STEP-4: If for all values of y belongs to B, the values of x obtained from x= g(y) are in A, then f is onto. If there are some y belongs B for which x, given by x = g(y), is not in A. Then, f is not onto.
INTO FUNCTION:
If f: A--> B is such that there are exists atleast one element in co-domain which is not the image of any element in domain, then f(x) is into.
**NOTE
1) If f is both and injective and surjective, then it is called Bijective mapping. The bijective functions are also named as invertible, nonsingular or biunform functions.
2) If a set A contains n distinct elements then the number of different functions defined from A-->A is nⁿ & out of it n! are one-one and rest are many-one.
3) f: R--> R is a polynomial
a) of even degree, then it will neither be Injective nor surjective.
b) of odd degree, then it will always be surjective, no general comment can be given on its injectivity.
EXERCISE -1
*************
1) A function f given as f: (2,7), (3,4), (7,9), (-1,6), (0,2),(5,3). Is this function
A) one-one.
B) into.
C) one-one onto.
D) one-one into.
2) interchange the above questions order of the elements in the ordered pairs and from the the new relation. Is this relation a
function
A) yes B) no C) cannot determined D) none
3) If above question is a function then it is
A) one-one
B) into
C) one-one onto
D) one-one into
4) To each person on the earth assign the number which corresponds to his age is
A) one-one
B) into
C) one-one onto
D) one-one into
E) none
5) To each country in the world assign the latitude and longitude of its capital is
A) one-one
B) into
C) one-one onto
D) one-one into
E) none
6) To each book written by only one author assign the author is
A) one-one
B) into
C) one-one onto
D) one-one into
E) none
7) To each country in the world which has a prime minister assign its Prime Minister is
A) one-one
B) into
C) one-one onto
D) one-one into
8) Let f: A-->B. Find f(A), i.e., the range of f, if is an onto function.
A) f(A)= B B) f(A)= A
C) f(A)= 1 D) f(A)= 1
9) the function f: R --> R given by f(x)= cos x for all x ∈ R, is
A) one-one
B) on to
C) either one-one or onto
D) neither one-one nor onto
10) let A={-1,1}. Let functions f, g and h of A onto A be defined by:
I) f(x)= x
II) g(x)= x³
III) h(x)= sin x,
which function, if any, is onto ?
A) I and II B) only I
C) Only II
D) neither I nor II
11) given A={2,3,4}, B{2,5,6,7}, construct an example of one-one mapping from A to B
A) {(2,5),(3,7),(4,6)}
B) {(2,2),(3,5),(4,2)}
C) {(2,5),(3,5),(4,6)}
D) {(2,5),(3,7),(4,7)}
12) given A={2,3,4}, B{2,5,6,7}, construct an example of one-one when A to B which is not one-one
A) {(2,2),(3,3),(4,4)}
B) {(2,3),(3,5),(4,3)}
C) {(2,2),(5,3),(6,4),(7,4)}
D) {(2,2),(3,5),(4,2)}
13) given A={2,3,4}, B{2,5,6,7}, construct an example of one-one when mapping from B to A
A) {(2,5),(3,7),(4,6)}
B) {(2,6),(3,7),(4,6)}
C) {(2,5),(3,5),(4,6)}
D) {(2,2),(5,3),(6,4),(7,4)}
14) Is the set of ordered pairs function ?
{(x,y): x is a person, y is the mother of x}
A) yes B) no C) can't be determined D) none
15) {(x,y): a is a person, b is an ancestor of a}
A) yes B) no C) can't be determined D) none
16) (x,y): x is a person, y is the mother of x} is
A) one-one function
B) onto function
C) into function
D) it is not a function
17) (x,y): a is a person, b is an ancestor of a} is
A) one-one function
B) onto function
C) into function
D) it is not a function
18) f: N--> N (N is the set of natural numbers) defined by
f(n)= 2n+3 for all n belongs to N is
A) one-one function
B) onto function
C) into function
D) it is not a function
19) let A={x= 0≤ x≤2} and B={1} then give an example of a function from A to B. can you define a function from B to A which is onto ?
20) the function f: R-> R, f(x)= x²+x is a
A) one-one function
B) many-one function
C) into function
D) many-one into function
21) Let A{1,2,3}, B={4,5,6,7} and f={(1,4),(2,5), (3,6)} be a function from A to B. than it is
A) one-one function
B) onto function
C) either one-one or onto function
D) neither one-one nor onto
E) one-one but not onto function
22) The function f: R-> R: f(x)= 3- 4x is
A) one-one function
B) into function
C) onto function
D) one-one onto or bijective function
23) If y= f(x)= (x+2)/(x-1), then x is
A) f(x) B) f(y) C) x D) y
24) f: N--> N given by f(x)= 5x is
A) Injective
B) surjective
C) Both A and B
D) bijective
E) one-one but not onto
25) Let f: R--> R given by f(x)= x⁴ is
A) one-one
B) not one-one
C) many to one
D) not many-one
26) On the set of Z, define f: Z--> Z as f(n)= n/2, n is even
0, n is odd
A) surjective but not injective
B) bijective
C) neither Injective nor surjective
D) injective but not surjective
EXERCISE -2
****************
1) which of the following functions from A to B are one-one and onto ?
a) f₁ ={(1,3),(2,5),(3,7)};
A={1,2,3}, B= {3,5,7}
b) f₂ = {(2,a),(3,b),(4,c);
A={2,3,4}, B= {a,b,c}
c) f₃={(a,x),(b,x),(c,z),(d,z)}
A={a,b,c,d}, B= {x,y, z}
2) Classify the following functions as injection, surjection or bijection:
a) f: N -> N given by f(x)= x²
b) f: Z -> Z given by f(x)= x²
c) f: N -> N given by f(x)= x³
d) f: Z -> Z given by f(x)= x³
e) f: R -> R, defined by f(x)= |x|.
f) f: Z -> Z, given by f(x)= x²+x
g) f: Z -> Z, defined by f(x)= x-5
h) f: R -> R, defined by f(x)= sinx
i) f: R -> R, defined by f(x)= x³+1
j) f: R -> R, defined by f(x)= x³- x
k) f: R -> R, defined by f(x)= sin x² + cos x².
l) f: Q -{3}-> Q, given by f(x)= (2x+3)/(x-3).
m) f: Q -> Q, defined by f(x)= x³+1
n) f: R -> R defined by f(x)= 5x³+4
o) f: R -> R defined by f(x)= 3-4x
p) f: R -> R given by f(x)= 1+x².
3) Give one example of a function:
a) which is one-one but not onto.
b) which is not one-one but onto.
c) which is neither one-one nor onto.
4) The function f: N -> N, defined by f(x)= 1+x²+ x is
A) one-one but not onto
B) one-one and onto
C) neither one-one nor onto
D) Either one-one nor onto
5) The function f: R -{3} --> R - {1} defined by f(x)= (x-1)/(x-3) is
A) bijection.
B) One-one
C) Onto
D) Into
6) Let A={-1,0,1} and f={x,x²): x ∈A}. Then f: A--> A is
A) neither one-one nor onto
B) Either one-one or onto
C) one-one but not onto
D) bijection
7) If f: A -> B, is an injection such that range of f(a), Determine the number of elements in A.
8) Let f: N -> N, defined by
f(x)= n+1, if n is odd
n-1, if n is even
Show that f is
A) a bijection
B) into
C) onto
D) one-one
9) Let A= {1,2,3}, write all one-one from A to itself.
10) the function f: R -> R, defined by f(x)= x - [x] is
A) either one-one or onto
B) neither one-one nor onto.
C) one-one but not onto
D) bijection
11) the signum function f: R -> R, defined by
f(x)= 1, if x > 0
0, if x= 0
-1, if x< 0 x
is
A) either one-one or onto
B) neither one-one nor onto. Find range of f
C) one-one or onto
D) bijection
12) The function f: R -> R, defined by f(x)= eˣ , is
A) one-one but not onto.
B) Either one-one or onto
C) neither one-one nor onto
D) bijection
13) from the previous question, What happens if the co-domain is replaced by R⁺₀(set of all positive real numbers).
14) The logarithmic function f: R⁺₀ R given by f(x)= logₐx, a> 0 is
A) one-one but not onto.
B) Either one-one or onto
C) neither one-one nor onto
D) bijection
15) If A={1,2,3}, then one-one function f: A--> A must be
A) onto
B) into
C) one-one
D) bijection
16) If A={1,2,3}, then a onto function f: A--> A must be
A) onto
B) into
C) one-one
D) bijection
17) Give examples of two one-one function f₁ and f₂ from R to R such that f₁ + f₂ : R --> R, defined by (f₁ + f₂)= f₁(x) + f₂(x) is not one-one.
18) Give examples of two surjective function f₁ and f₂ from Z to Z such that f₁ + f₂ is not surjective.
19) If f₁ and f₂ are one-one maps from R to R, then the product f₁ x f₂ : R --> R defined by (f₁ x f₂)= f₁(x) f₂(x) need not be
A) one-one
B) into
C) onto
D) bijection
20) Find the number of all onto function from the set A={1,2,3,......n} to itself.
EXERCISE-3
-------------------
1) Given that f: x--> 3x-2 and g: x--> x⁵ for all of x ∈ R, find
a) g(f)(x). (3x-2)⁵
b) f(f))(x). 9x -8
c) f(g(x)+ 3). 3x⁵+7
d) f(g(x) - f(x)). 3x⁵ - 9x +4
2) If f(x)= x²-1 and g(x)= √x, find
A) (fog)(x). x-1
B) (gof)(x). √(x²-1)
C) (fof)(x). (x²-1)² - 1
D) (gog)(x). ⁴√x
3) If f(x)= 4x² +1 and g(x)= 1/(x+2) find
A) (gof)(2). 1/19
B) (fog)(-1). 5
C) (gof)(x). 1/(4x²+3)
D) (fog)(x). 4/(x+2)² + 1
4) given f and g defined by f(x)= x² + 2 and g(x)= 1 - 1/x, find the composite of
A) (gof). (x²+1)/(x²+2)
B) fog). (3x²-2x +1)/x²
5) Find the composite g(f) when g(x)= | x | and f(x)= x²- 3x +1.
6) If h(x)= sec x, find (hoh)² at x=π/4. sec²(√2)
7) If f(x)= log {(1+x)/(1-x)} and g(x) = (3x+x³)/(3x²+1). Then f{g(x)} is
A) - f(x) B) -3f(x) C) 3f(x). D){f(x)}³
8) If f: R --> R, g: R-->R and functions defined by f(x)= 3x-2, g(x)= x²+1, find gof.
9) If f: R --> R, g: R-->R and functions defined by f(x)= 3x-2 is
A) one-one function
B) onto function
C) into function
D) one-one and onto function
10) If f(x)= (x+1)/(x-1), then find (fofofof) is
A) 1/x B) x C) x² D) indeterminate
11) If f: x-->2x, g: x--> x² and h: x--> x+1, find
A) ho(gof). 4x²+1
B) (hog)of. 4x²+1
What do you notice? Associative
12) If f: R-->R, g: R--> R are defined by f(x)= 2x+3, and g(x)= x²+7, then find the values of x for which f(g(x))= 25. ±2
13) If f be a greatest integer function and g be an absolute value function, find the value of (fog)(-3/2)+ (gof)(4/3) 2
14) Let f, g and h be function from R to R, Show that (f+g)oh - foh + goh.
15) If f(x)= 3x-2 and g(x)= x², find
A) (gof)(3). 49
B) (fog)(1). 1
C) (fof)(0). 8
16) If f(x)= |x+2| and g(x)= -x², find
A) (gof)(3). -25
B) (fog)(1). 1
C) (fof)(0). 4
17) If f(x)= x² -1 and g(x)= √x, find
A) (gof)(3). √8
B) (fog)(1). 0
C) (fof)(0). 0
18) If f(x)= x +5 and g(x)= x²-3, find
A) f(g(0)). 2
B) g(f(0)). 22
C) f(g(x)). x²+2
D) g(f(x)). x²+10x+22
E) f(f(x)). x+10
F) g(g(x)). x⁴ - 6x²+6
G) f(f(-5)). 5
H) g(g(2)). . -2
19) If u(x)= 4x -5 and v(x)= x², and f(x)= 1/x, find
A) u(v(f(x))). 4/x² - 5
B) u(f(v(x))). 4/x² - 5
C) f(u(v(x))). 1/(4x²-5)
20) If g(t)= t² -1 and f(x)= x +1 find
A) g(f(0))+ f(g(0)). 1
B) g(f(2) +3). 30
21) If f(2x+1)/(3x -2), then (fof)(2) is equal to
A) 1 B) 2 C) 3 D) 4
22) If f(x)= (1-x)/(1+x), then f(f(cos 2m)) is equal to
A) cos 2m B) tan 2m
C) sec 2m D) cot 2m.
23) If f(x)= sin²x and the composite function g[f(x)]=| sin x|
Then the function g(x) is equal to
A) -√x B) √x C) √(x-1) D) √(x+1)
24) If f: R-->R and g: R-->R are defined respectively as f(x)= x² + 3x +1 and g(x)= 2x-3, find
A) fog.
B) gof
25) If f: R-->R, f(x)= x², g: R--> R, g(x)= cos x, find
A) fog.
B) gof
And show that fog≠ gof.
26) If f(x)= log x, g(x) = x³, then prove that f(g(a))+ f(g(b))= 3f(ab)
27) If f(x)= ⁴√(25 - x⁴) for 0 <x< √5, then prove that f[f(1/2)]=1/2
28) If f: R-->R is defined by f(x)= 3x+2, find f(f(x)). 9x+8
29) If f: R-->R is defined by f(x)= ³√(3-x), then find fof(x). x
30) If the function f: R-->R is defined by f(x)= (x+3)/3 and g : R-->R is defined by g(x)= 2x-3, find
A) fog. x
B) gof. x
31) If f(x)= x/(x-1), then show that f(a)/f(a+1)= f(a²).
EXERCISE-4
---------------------
1) When f: R --> and g: R--> R are defined by f(x)=2x+3, g(x)=x²+5. Find
a) gof. 4x²+12x+14
b) fog. 2x²+13
2) When f: R --> and g: R--> R are defined by f(x)=2x+x² , g(x)=x³. Find
A) gof. (x²+2x)³
B) fog. 2x³+ x⁶
3) When f: R --> and g: R--> R are defined by f(x)=x²+8, g(x)= 3x³+1 find
A) gof. 3(x²+8)³+1
B) fog. 9x⁶+ 6x³+9
4) Let f={(1,-1),(4,-2),(9,-3),(16,4)} and g={(-1,-2),(-2,-4),(-3,-6),(4,8)}. Show that gof is defined while fog is not defined. Also, find gof. {(1,-2),(4,-4),(9,-6),(16,8)}
5) Let f={(3,1),(9,3),(12,4)} and g={(1,3),(3,3),(4,9),(5,9)}. Show that gof and fog are both defined. Also, find fog and gof.
gof={(3,3),(9,3),(12,9)}
fog={(1,1),(3,1),(4,3),(5,3)}
6) Verify associativity for the following three mappings: f: N-->Z₀ (the set of non-zero integers), g: Z₀ --> O and h: Q --> R given by f(x)= 2x, g(x)= 1/x and h(x)= eˣ.
7) Let A={a, b, c}, B={u, v, w} and let f and g be two functions from A to B and from B to A respectively defined as:
f={(a,v),(b,u),(c,w), g={(u,b),(v,a),(w,c)}.
Show that f and g both are bijections and find fog and gof.
fog={(u,v),(v,v),(w,w)}
gof={(a,a),(b,b),(c,c)}
8) Find fog(2) and gof(1) when: f: R --> R ; f(x)= x²+8 and g: R--> R; g(x)= 3x³ +1. 633, 2188
9) Let R⁺ be the set of all non-negative real numbers. If f: R⁺ --> R⁺ and g: R⁺ --> R⁺ are defined as f(x)= x² and g(x)= + √x.
Find fog and gof. Are they equal functions. fog(x)=x, gof(x)=x
10) If f: A--> B and g: B--> C are one-one functions, show that gof is a one-one function.
11) If f: A--> B and g: B--> C are onto functions show that gof is an onto function.
12) Let f: R--> R and g: R--> R be defined by f(x)= x² and g(x)= x+1. show that fog≠ gof.
13) Let f: R--> R and g: R--> R be defined by f(x)= x +1 and g(x)= x -1. show that fog= gof= Iᵣ.
14) Consider f: N--> N, and g: N--> N and h: N--> R defined by f(x)= 2x, and g(y)= 3y+4 and h(z)= sinz for all x,y , z belongs to N. show that ho(gof)= (hog)of.
15) If f: R--{7/5} --> R --{3)5} be defined as f(x)= (3x+4)/(5x-7) and g: R--{3/5} --> R--{7/5} be defined as g(x)= (7x+4)/(5x-3). Show that gof = Iₐ and fog= Iₘ. Where M= R--{3/5} and A= R --{7/5}.
16) If f: R--> R is defined by f(x)= x² - 3x+2, find f(f(x)). x⁴-6x³+10x²3x
17) Let A={x∈ R: 0≤ x ≤1}. If f: A-->A is defined by
f(x)= x, if x ∈ Q
1- x, if x not belongs to Q
Then prove fof(x)= x for all x∈ A.
18) Let f: R--> R and g: R--R be two functions such that fog(x)= sin x² and gof(x)= sin²x. Then, find f(x), g(x). sinx, x²
19) If f: R--> R be given by f(x)= sin²x + sin²(x+π/3) + coax vos(x+π/3) for all x x∈R and g: R--> R be such that g(5/4)= 1, then prove that gof R--> R is a constant function.
20) Let Z--> Z be defined by f(n) = 3n for all n belongs to Z and g: Z--> Z be defined by
g(n) = n/3, n is a multiple of 3
0, if n is not a multiple of 3 for all n belongs to Z
Show that gof = I(z) and fog≠ I(Z).
21) Let f: R--> R be a function given by f(x)= ax+ b for all x∈ R. Find the constants aand b such that fof = I(R).
22) Let f: Z--> Z be
COMPOSITION OF REAL FUNCTIONS:::
EXERCISE -5
_______________
1) Let f(x)= x²+x+1 and. g(x)= sinx. Show that fog≠ gof.
2) Let f(x)= 1+x, 0≤ x≤ 2
3 - x, 2< x≤ 3
Find fof. 2+x, if, 0≤x≤1
2 -x, if, 1<x≤2
4- x, if, 2< x≤ 3
3) If f(x)=2x+5 and g(x)=x²+1 be two real functions, then describe each of the following functions:
A) fog. R-->R is given by fog(x)= 2x²+7
B) gof. R-->R is given by gof(x)= 4x²+20x+26
C) fof. R-->R is given by fof(x)= 4x+5
D) f². R-->R is given by f²(x)= 4x²+20x+25
Also show that fof≠ f²
4) If f(x)= |x|, prove that fof= f
5) If f(x)= (a - xⁿ)¹⁾ⁿ, where a> 0 and n belongs to N. Show that f(f(x)) = x.
6) If f(x)= sinx and g(x)= 2x be two real functions, then describe each of the following functions:
A) gof. R-->R is given by gof(x)= 2sinx
B) fog. R-->R is given by fog(x)= sin 2x.
Are these equal functions? No
7) If f(x)= √(x+3) and g(x)= x²+1 be two real function, then find
A) fog. fog: R-->R is given by fog(x)= √(x² +4)
B) gof. gof: [-3, ∞) -->R is given by gof(x)= x+4
8) let f be a real function given by f(x)= √(x-2) find each of the following:
A) fof. fof: [6, ∞) -->R is given by fof(x)= √{√(x-2) -2}
B) fofof. fofof: [38, ∞) -->R is given by (fofof)(x)= √[√{√(x-2) -2} -2]
C) (fofof)(38). 0
D) f². f²[2, ∞) -->R is given by f²(x)= x-2
9) If f: (-π/2, π/2) --> R and g: [-1,1] --> R be defined as f(x)= tanx and g(x)= √(1-x²) respectively. Describe
A) fog. fog: [-1,1] --> R defined as fog(x)= tan√(1-x²).
B) gof. gof: (-π/4, π/4) --> R is defined as gof(x) = √(1- tan²x).
10) If f(x)= √(1-x) and g(x)= log x are two real functions then describe functions
A) fog. fog: (0,e] --> R is given by (fog)(x)=√(1- log x)
B) gof. gof: (-∞,1)-->R is given by (gof)(x)= 1/2 log(1-x)
11) let f be any real function and let g be a function given by g(x)= 2x. prove that gof= fof.
12) Let f, g, h be real functions given by f(x)= sin x, g(x)= 2x and h(x)= cos x. prove that fog = go(fh).
13) Find fog and gof, if
A) f(x)= eˣ, g(x)= logₑx.
fog: (0, ∞) -->R is given by fog(x)= x
gof: R -->R is given by gof(x)= x
B) f(x)= x², g(x)= cosx
fog: R -->R is given by fog(x)= cos²x
gof: R -->R is given by gof(x)= cosx²
C) f(x)= |x|, g(x)= sinx.
fog: R -->R is given by fog(x)= |sinx|.
gof: R -->R is given by gof(x)=sin |x|
D) f(x)= x+1, g(x)= eˣ.
fog: R -->R is given by fog(x)=eˣ +1
gof: R -->R is given by gof(x)=eˣ⁺¹
E) f(x)= sin⁻¹x, g(x)= x².
fog: [-1,1] -->R is given by fog(x)= sin⁻¹(x²)
gof: [-1,1] -->R is given by fog(x)= (sin⁻¹x)².
F) f(x)= x+1, g(x)= sinx.
fog:R -->R is given by fog(x)= sinx +1
gof: R -->R is given by gof(x)= sin (x+1)
G) f(x)= x+1, g(x)= 2x+3.
fog: R-->R is given by fog(x)= 2x+4
gof: R-->R is given by gof(x)= 2x+5.
H) f(x)= c, c belongs to R, g(x)= sin x².
fog: R -->R is given by fog(x)= c
gof:R -->R is given by gof(x)= sin c²
I) f(x)= x²+2, g(x)= 1- 1/(1-x).
fog: {1}-->R is given by fog(x)= (3x²- 4x+2)/(1-x)²
gof: R-->R is given by gof(x)= (x²+2)/(x²+1).
⁻¹∞ ∈ ∧∨ ⇒ ⇔ ∪ ∩ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉ ₀
INVERSE OF A FUNCTION
Let f: A --> B be a bijection. Then a function g: B--> A which associates each element y ∈B to a Unique elements x ∈ A such that f(x)= y is called the inverse of f.
The inverse of f generally denoted by f⁻¹
Thus, if A--> B is a bijection, then f⁻¹ : B --> is such that
f(x)= y⇔ f⁻¹(y)= x
In order to find the inverse surface of a bijection, we follow the following steps.
Let f: A--> B be a bijection. To find the inverse of we follow the following steps:
step 1. Put f(x)= y, where y ∈B and x ∈A.
Step 2. solve f(x)=y to obtain x in terms of y.
Step 3. In the relation obtain in step 2 replace x by f⁻¹(y) to obtain the required inverse of f.
PROPERTIES OF INVERSE OF A FUNCTION
A: The inverse of a bijection is unique.
B: The inverse of a bijection is also a bijection.
C: If f: A--> B is a bijection and g: B--> is the inverse of f, then fog= Iᴮ and gof = Iᴬ, where Iᴬ and Iᴮ are the identity functions on the sets A and B respectively.
D: If f: A--> B and g: B--> C are two bijections, then gof: A --> C is a bijection and (gof)⁻¹ = f⁻¹og⁻¹.
E: Let f: A--> B and g: B---> A be two functions such that gof = Iᴬ and fog = Iᴮ then, f and g are bijection and g = f⁻¹.
F: Let f: A--> B be an invertible function. Show that the inverse of f⁻¹ is f, i.e., (f⁻¹)⁻¹ = f.
Note 1: sometimes f: A--> B is one one but not onto. In such a case f is not invertible. But f: A--> Range (f) is both one and onto. So, it is invertible and its iniverse can be found.
EXERCISE-A
1) if A= {1,2,3,4}, B={2,4,6,8} and A-->B is given by f(x)= 2x, then write f and f⁻¹ as a set of ordered pairs.
2) Let S={1,2,3}. Determine whether the function f: S--> S defined as below have inverse. Find f⁻¹ , if it is exists.
i) f={(1,1),(2,2),(3,3)}
ii) f= {(1,2),(2,1),(3,1)}
iii) f ={(1,3),(3,2), (2,1)}
3) Consider f: {1,2,3}--> {a,b,c}given by f(1)= a, f(2)= b and f(3)=c. Find the iniverse (f⁻¹)⁻¹ of f⁻¹ . Show that (f⁻¹)⁻¹ = f.
4) If f: R--> R is defined by f(x)= 2x + 7. Prove that f is a bijection. Also, find the inverse of f.
5) If f: R--> R is a bijection given by f(x)= x³+ 3, find f⁻¹(x).
6)et f: R--> R be defined by f(x)= 3x -7. Show that f is invertible and hence find f⁻¹.
7) Show that f: R -{0} --> given by f(x)= 3/x is invertible and is inverse of itself.
8) Let f: N U {0} --> N U {0} be defined by
f(n)= n+1, if n is even
n -1, if n is odd
9) Show that f:[-1,1] --> R, given by f(x)= x/(x+2) is one-one. Find the inverse of the function f: [-1,1] --> Range (f).
10) If A={1,2,3,4} and B={a,b,c,d}. Define any four bijection from A to B. Also give their inverse functions.
11) Find f⁻¹ if it is exists: f: A--> B where
i) A={0,-1,-3,2}; B={-9,-3,0,6} and f(x)=3x.
ii)A={1,3,5,7 ,9}; B={0,1, 9,25, 49,81} and f(x)= x².
12) If f: R--> R be defined by f(x)= x³ -3, then prove that f⁻¹ exist and find a formula for f⁻¹. Hence, find f⁻¹(24) and f⁻¹(5). ³√(3+x), 3, 2
13) Let A{1,2,3,4}, B{3,5,7,9}, C{7,23,47,79} and f: A---> B, g: B--> C be defined as f(x)= 2x+1 and g(x)= x²- 2. Express (gof⁻¹) and f⁻¹o g⁻¹ as the sets of ordered pairs and verify that (gof)⁻¹= f⁻¹og⁻¹.
14) A function f: R--> R is defined as f(x)= x³+ 4. Is it is a bijection or not ? In case it is a bijection, find f⁻¹(3). -1
15) If f: A--> A, g: A--> A are two bijections, then prove that
A) fog is an injection
B) fog is a surjection
16) If f: Q--> Q, g: Q--> Q are two functions defined by f(x)= 2x and g (x)= x+2. Show that f and g are bijective maps. Verify that (gof)⁻¹= f⁻¹og⁻¹.
17) Let f be defined from R to R such that f(x)= cos(x+2). Is f invertible? Justify your answer. Not invertible
18) let f: [-1, ∞]--> [-1, ∞] is given by f(x)= (x+1)² -1, x≥ -1. Show that f is invertible.
19) If f(x)= (4x+3)/(6x-4), x≠ 2/3, show that fof(x)= x for all x≠ 2/3. What is inverse of f. f⁻¹(x)= (4x+3)/(6x -4)
20) Consider f: R--> given by f(x)= 4x + 3, Show that f is invertible. Find the inverse of f. f⁻¹(x)= (x- 3)/4
21) Consider f: R-->R -->[4,∞) given by f(x)= x² + 4, Show that f is invertible with f⁻¹ of f given by f⁻¹(x)= √(x- 4), where R is the set of all non-negative real numbers.
22) Consider f: R-->[-5,∞) given by f(x)= 9x² + 6x -5, Show that f is invertible with f⁻¹(x)= {√(x+6) -1}/3.
23) State with reason whether the following functions have inverse:
i) f:{1,2,3,4}--> {10} with f={(1,10),(2,10), (3,10),(4,10)}. No, f is many to one
ii) g:{5, 6, 7, 8}-->{1, 2, 3, 4} with g {(5,4),( 6, 3), (7, 4), (8, 2)}. No, g is many one
iii) h:{2,3, 4, 5}-->{7, 9, 11, 13} with h={(2,7),(3,9),(4 ,11),(5,13)}. Yes, h is a bijection
24) Show that the function f: Q--> Q defined by f(x)= 3x+ 5 is invertible. Also, find f⁻¹ . f⁻¹(x)= (x-5)/3
25) If f: R-->(-1,1) defined by f(x)= (10ˣ -10⁻ˣ)/(10ˣ+ 10⁻ˣ) is invertible, find f⁻¹. f⁻¹(x)= 1/2 log{(1+x)/(1- x)}
26) If f: R-->(0,2) defined by f(x)= (eˣ - e⁻ˣ)/(eˣ+ e⁻ˣ) + 1 is invertible, find f⁻¹. f⁻¹(x)= 1/2 log{x/(2- x)}
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