Advertisements
Advertisements
प्रश्न
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
उत्तर १
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)`
उत्तर २
`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)`
उत्तर ३
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.
संबंधित प्रश्न
Let A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is ______.
Test whether the following relation R1 is (i) reflexive (ii) symmetric and (iii) transitive :
R1 on Q0 defined by (a, b) ∈ R1 ⇔ a = 1/b.
Show that the relation R on 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.
Let Z be the set of all integers and Z0 be the set of all non-zero integers. Let a relation R on Z × Z0be defined as (a, b) R (c, d) ⇔ ad = bc for all (a, b), (c, d) ∈ Z × Z0,
Prove that R is an equivalence relation on Z × Z0.
If R = {(x, y) : x2 + y2 ≤ 4; x, y ∈ Z} is a relation on Z, write the domain of R.
Let R be the equivalence relation on the set Z of the integers given by R = { (a, b) : 2 divides a - b }.
Write the equivalence class [0].
The relation 'R' in N × N such that
(a, b) R (c, d) ⇔ a + d = b + c is ______________ .
Let R be a relation on N defined by x + 2y = 8. The domain of R is _______________ .
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
Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of the following:
a mapping from B to A
The following defines a relation on N:
x + 4y = 10 x, y ∈ N.
Determine which of the above relations are reflexive, symmetric and transitive.
Consider the non-empty set consisting of children in a family and a relation R defined as aRb if a is brother of b. 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 A = {1, 2, 3}, then the domain of the relation R = {(1, 1), (2, 3), (2, 1)} defined on A is ____________.
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 ____________.
Let R be the relation “is congruent to” on the set of all triangles in a plane is ____________.
A relation R in set A = {1, 2, 3} is defined as R = {(1, 1), (1, 2), (2, 2), (3, 3)}. Which of the following ordered pair in R shall be removed to make it an equivalence relation in A?
Given set A = {1, 2, 3} and a relation R = {(1, 2), (2, 1)}, the relation R will be ____________.
Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.
Answer the following using the above information.
- Let relation R be defined by R = {(L1, L2): L1║L2 where L1, L2 ∈ L} then R is ____________ relation.
Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.
Answer the following using the above information.
- Let R = {(L1, L2 ): L1 is parallel to L2 and L1: y = x – 4} then which of the following can be taken as L2?
The value of k for which the system of equations x + ky + 3z = 0, 4x + 3y + kz = 0, 2x + y + 2z = 0 has nontrivial solution is
Which one of the following relations on the set of real numbers R is an equivalence relation?
Which of the following is/are example of symmetric
Let a set A = A1 ∪ A2 ∪ ... ∪ Ak, where Ai ∩ Aj = Φ for i ≠ j, 1 ≤ i, j ≤ k. Define the relation R from A to A by R = {(x, y): y ∈ Ai if and only if x ∈ Ai, 1 ≤ i ≤ k}. Then, R is ______.
Let N be the set of all natural numbers and R be a relation on N × N defined by (a, b) R (c, d) `⇔` ad = bc for all (a, b), (c, d) ∈ N × N. Show that R is an equivalence relation on N × N. Also, find the equivalence class of (2, 6), i.e., [(2, 6)].
A relation R on (1, 2, 3) is given by R = {(1, 1), (2, 2), (1, 2), (3, 3), (2, 3)}. Then the relation R is ______.
Statement 1: The intersection of two equivalence relations is always an equivalence relation.
Statement 2: The Union of two equivalence relations is always an equivalence relation.
Which one of the following is correct?
If a relation R on the set {a, b, c} defined by R = {(b, b)}, then classify the relation.
When is a relation called an equivalence relation?
