Advertisements
Advertisements
Questions
If `f(x) = (4x + 3)/(6x - 4), x ≠ 2/3`, show that fof (x) = x for all `x ≠ 2/3`. Also, find the inverse of f.
If `f(x) = (4x + 3)/(6x - 4), x ≠ 2/3`, then show that (fof) (x) = x, for all `x ≠ 2/3`. Also, write inverse of f.
Advertisements
Solution 1
f (x) = `(4x + 3)/(6x - 4) `
`f (f (x)) = (4 f(x) + 3)/(6 f(x) - 4)`
`f(f(x))= (4 ((4x + 3)/(6x - 4))+3)/(6((4x + 3)/(6x - 4))-4)`
` fof (x) = (((16x + 12 + 18x - 12)/(6x -4)))/(((24x + 18 - 24 x + 16)/(6x - 4)))`
` fof (x) = (34x)/34`
fof (x) = x
For inversere y = `(4x + 3)/(6x - 4)`
6xy – 4y = 4x + 3
6 xy – 4x = 4y + 3
x(6y – 4) = 4y + 3
`x = (4y + 3)/(6y - 4) ⇒ y = (4x + 3)/(6x - 4)`
`⇒ f^(-1) (x) = (4x + 3)/(6x - 4)`
Solution 2
`f(x) = (4x +3)/(6x -4) x ≠ 2/3`
`f "of"(x) = (4((4x +3)/(6x - 4))+ 3)/(6((4x +3)/(6x - 4)) - 4)`
= `(16x + 12 + 18x - 12)/(24x + 18 - 24x + 16)`
= `(34x)/(34) = x`
Therefore, fof (x) = x, for all `x ≠ 2/3`
⇒ fof = I
Hence, the given function f is invertible and the inverse of f is itself.
`y = (4x + 3)/(6x - 4)`
`6xy - 4y = 4x +3`
`6xy - 4y = 4y +3`
`x = (4y + 3)/(6y -4)`
∴ `f(x) = (4x +3)/(6x - 4)`
Solution 3
fof (x) = f(f(x))
= `f((4x + 3)/(6x - 4))`
= `(4((4x + 3)/(6x - 4)) + 3)/(6((4x + 3)/(6x - 4)) - 4)`
= `(16x + 12 + 18x - 12)/(24x + 18 - 24x + 16)`
= `(34x)/34`
= x
Now, suppose `y = (4x + 3)/(6x - 4)`
⇒ 6xy – 4y = 4x + 3
⇒ 6xy – 4x = 3 + 4y
⇒ x(6y – 4) = 3 + 4y
⇒ `x = (3 + 4y)/(6y - 4)`
Therefore, `f^-1 = (3 + 4y)/(6y - 4)`
So here inverse of f is equal to function f.
Notes
Students should refer to the answer according to their questions.
RELATED QUESTIONS
Show that the relation R in the set R of real numbers, defined as R = {(a, b) : a ≤ b2} is neither reflexive nor symmetric nor transitive.
Show that the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12} given by R = {(a, b) : a = b} is an equivalence relation. Find the set of all elements related to 1.
Given an example of a relation. Which is symmetric and transitive but not reflexive.
Let L be the set of all lines in the XY plane and R be the relation in L defined as R = {(L1, L2) : L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.
Let A = {x ∈ Z : 0 ≤ x ≤ 12}. Show that R = {(a, b) : a, b ∈ A, |a – b| is divisible by 4}is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [2]
The following relation is defined on the set of real numbers. aRb if |a| ≤ b
Find whether relation is reflexive, symmetric or transitive.
If A = {1, 2, 3, 4} define relations on A which have properties of being symmetric but neither reflexive nor transitive ?
Give an example of a relation which is symmetric and transitive but not reflexive?
Let Z be the set of integers. Show that the relation
R = {(a, b) : a, b ∈ Z and a + b is even}
is an equivalence relation on Z.
If R and S are relations on a set A, then prove that R is reflexive and S is any relation ⇒ R ∪ S is reflexive ?
If a relation R is defined on the set Z of integers as follows:
(a, b) ∈ R ⇔ a2 + b2 = 25. Then, domain (R) is ___________
Mark the correct alternative in the following question:
The maximum number of equivalence relations on the set A = {1, 2, 3} is _______________ .
Mark the correct alternative in the following question:
Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to b for all a, b T. Then, R is ____________ .
If A = {a, b, c}, B = (x , y} find A × B.
Consider the set A = {1, 2, 3} and R be the smallest equivalence relation on A, then R = ______
If A = {1, 2, 3, 4 }, define relations on A which have properties of being:
reflexive, symmetric and transitive
Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of the following:
a mapping from A to B which is not injective
Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to b ∀ a, b ∈ T. Then R is ______.
An integer m is said to be related to another integer n if m is a integral multiple of n. This relation in Z is reflexive, symmetric and transitive.
Let R be the relation on N defined as by x + 2 y = 8 The domain of R is ____________.
Which of the following is not an equivalence relation on I, the set of integers: x, y
Let R be the relation “is congruent to” on the set of all triangles in a plane is ____________.
Given set A = {1, 2, 3} and a relation R = {(1, 2), (2, 1)}, the relation R will be ____________.
Given set A = {a, b, c}. An identity relation in set A is ____________.
An organization conducted a bike race under 2 different categories-boys and girls. Totally there were 250 participants. Among all of them finally, three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project. Let B = {b1,b2,b3} G={g1,g2} where B represents the set of boys selected and G the set of girls who were selected for the final race.
Ravi decides to explore these sets for various types of relations and functions.
- Ravi wishes to form all the relations possible from B to G. How many such relations are possible?
If f(x + 2a) = f(x – 2a), then f(x) is:
Let R1 and R2 be two relations defined as follows :
R1 = {(a, b) ∈ R2 : a2 + b2 ∈ Q} and
R2 = {(a, b) ∈ R2 : a2 + b2 ∉ Q}, where Q is the set of all rational numbers. Then ______
Let R = {(x, y) : x, y ∈ N and x2 – 4xy + 3y2 = 0}, where N is the set of all natural numbers. Then the relation R is ______.
