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
Given an example of a relation. Which is symmetric but neither reflexive nor transitive.
Given an example of a relation. Which is reflexive and transitive but not symmetric.
Let A = {1, 2, 3}, and let R1 = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)}, R2 = {(2, 2), (3, 1), (1, 3)}, R3 = {(1, 3), (3, 3)}. Find whether or not each of the relations R1, R2, R3 on A is (i) reflexive (ii) symmetric (iii) transitive.
The following relation is defined on the set of real numbers.
aRb if a – b > 0
Find whether relation is reflexive, symmetric or transitive.
Give an example of a relation which is reflexive and transitive but not symmetric?
Give an example of a relation which is symmetric but neither reflexive nor transitive?
Given the relation R = {(1, 2), (2, 3)} on the set A = {1, 2, 3}, add a minimum number of ordered pairs so that the enlarged relation is symmeteric, transitive and reflexive.
If R and S are relations on a set A, then prove that R and S are symmetric ⇒ R ∩ S and R ∪ S are symmetric ?
Write the smallest equivalence relation on the set A = {1, 2, 3} ?
Let R = {(a, a), (b, b), (c, c), (a, b)} be a relation on set A = a, b, c. Then, R is _______________ .
Let R be the relation on the set A = {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Then, _____________________ .
Let A = {1, 2, 3}. Then, the number of equivalence relations containing (1, 2) is ______.
The relation R = {(1, 1), (2, 2), (3, 3)} on the set {1, 2, 3} is ___________________ .
S is a relation over the set R of all real numbers and it is given by (a, b) ∈ S ⇔ ab ≥ 0. Then, S is _______________ .
Show that the relation R on R defined as R = {(a, b): a ≤ b}, is reflexive, and transitive but not symmetric.
Show that the relation S in the set A = [x ∈ Z : 0 ≤ x ≤ 12] given by S = [(a, b) : a, b ∈ Z, ∣a − b∣ is divisible by 3] is an equivalence relation.
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 the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by R = {(a, b) : |a – b| is a multiple of 4}. Then [1], the equivalence class containing 1, is:
Given set A = {a, b, c}. An identity relation in set A is ____________.
A general election of Lok Sabha is a gigantic exercise. About 911 million people were eligible to vote and voter turnout was about 67%, the highest ever
Let I be the set of all citizens of India who were eligible to exercise their voting right in the general election held in 2019. A relation ‘R’ is defined on I as follows:
R = {(V1, V2) ∶ V1, V2 ∈ I and both use their voting right in the general election - 2019}
- The above-defined relation R is ____________.
Sherlin and Danju are playing Ludo at home during Covid-19. While rolling the dice, Sherlin’s sister Raji observed and noted the possible outcomes of the throw every time belongs to set {1,2,3,4,5,6}. Let A be the set of players while B be the set of all possible outcomes.
A = {S, D}, B = {1,2,3,4,5,6}
- Let R ∶ B → B be defined by R = {(x, y): y is divisible by x} is ____________.
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.
Find: `int (x + 1)/((x^2 + 1)x) dx`
Define the relation R in the set N × N as follows:
For (a, b), (c, d) ∈ N × N, (a, b) R (c, d) if ad = bc. Prove that R is an equivalence relation in N × N.
Read the following passage:
|
An organization conducted bike race under two different categories – Boys and Girls. There were 28 participants in all. 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. |
Based on the above information, answer the following questions:
- How many relations are possible from B to G? (1)
- Among all the possible relations from B to G, how many functions can be formed from B to G? (1)
- Let R : B `rightarrow` B be defined by R = {(x, y) : x and y are students of the same sex}. Check if R is an equivalence relation. (2)
OR
A function f : B `rightarrow` G be defined by f = {(b1, g1), (b2, g2), (b3, g1)}. Check if f is bijective. Justify your answer. (2)
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 ______.
If a relation R on the set {a, b, c} defined by R = {(b, b)}, then classify the relation.
