TextBook Solutions
- Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f. f : R → R given by f(x) = 4x + 3. f(x1) = 4x1 + 3, f(x2) = 4x2 + 3 If f(x1) = f(x2), then 4x1 + 3 = 4x2 + 3 or 4x1 = 4x2 ⇒ x1 = x2. ⇒ f is o... 1
- Consider f : R+ → [4, ∝) given by f(x) = x² + 4. Show that f is invertible with the inverse of f-1 of f f(x1) = x²1 + 4 and f(x2) = x²2 + 4 So, f(x1) = f(x2) ⇒ x²1 + 4 = x²2 + 4 or x²1 = x²2 ⇒ x1 = x2 As x ∈ R, ∴ x > 0, x²1 = x²2 ⇒ x1 = x2 ⇒ f... 1
- Let f : X → Y be an invertible function. Show that f has a unique inverse. If f is invertible, then gof(x) = Ix and fog(y) = Iy. ⇒ f is one-one and onto. Let there be two inverses g1 and g2 ∴ fog1 (y)= Iy, f0g2(y) =... 1
- Consider f : {1, 2, 3} → {a, b, c] given by f(1) = a, f(2) – b, f(3) = c. Find f-1 and show that (f-1)-1 = f. f : {1, 2, 3} → {a, b, c} so that f(1) = a, f(2) = b, f(3) = c. Now, let X = {1, 2, 3}, Y = {a, b, c). ∴ f : X → Y ∴ f-1 : Y → X such that f... 1
- Let f : X → Y be an invertible function. Show that the inverse of f-1 is f, i.e., (f-1)-1 = f. f : X → Y is an invertible function. ∴ f is one-one and onto ⇒g : Y → X, where g is also one-one and onto such that gof(x) = Ix and fog(y) =... 1
- If f : R → R be given by f(x) = (3-x³)\(\left(3-x^{3}\right)^{\frac{1}{3}}\), when fof(x) is Correct answer is (C) x 2
- Let f : R – (- \(\frac { 4 }{ 3 }\)) → R be a function, defined as f(x) = \(\frac { 4x }{ 3x+4 }\) f : R – -4 / 3 → R is a function defined by f(x) = 4x / 3x+4 Let y = 4x / 3x+4 . ⇒ 3xy + 4y = 4x or x(4 – 3y) = 4y ∴ x = 4y / 4-3y ∴ f-1(4y)... 1
- Show that the function f: R → R, defined by f (x) = \(\frac{1}{x}\) is one-one onto (i) f(x) = 1/x , If f(x1) = f(x2) ⇒ frac1x₁ = frac1x₂ ⇒ x1 = x2 Each x ∈ R, has a unique image in codomain. ⇒ f is one-one. (ii) For each y... 1
- Check the injectivity and surjectivity of the following functions: (i) f : N → N is given by (i) f : N → N given by f(x) = x2. (a) f(x1) = f(x2) ⇒ x₁² = x₂² ⇒ x1 = x2. ∴ f is one-one, i.e., it is injective. (b) There are some members... 1
- Prove that greatest integer function f : R → R, given by f(x) = [x], is neither one-one nor onto f : R → R given by f(x) = [x]. (a) f(1.2) = 1, f(1.5) = 1 ⇒ f is not one-one. (b) All the images of x ∈ R belonging to its domain have integ... 1
- Show that the Modulus Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where | x | is x f : R → R given by f(x) = | x |. (a) f(-1) = |-1| = 1, f(1) = |1| = 1. ⇒ – 1 and 1 have the same image. ∴ f is not one-one. (b) No negative... 1
- Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {1, 4), (2, 5), (3, 6)} be a function from A to B. A = (1, 2, 3}, B = {4, 5, 6, 7} and f = {(1, 4), (2, 5), (3, 6)}. Every member of A has a unique image in B as shown in the figure. ∴ f is o... 1
- In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer. (i) f : R → R defined by f(x) = 3 – 4x. f(x1) = 3 – 4x1, f(x2) = 3 – 4x2 (a) f(x1) = f(x2) ⇒ 3 – 4x1 = 3 – 4x2 ∴ x1 = x2 This shows that f i... 1
- Let A and B be two sets. Show that f : A × B → B × A such that f(a, b) = (b, a) is bijective function. We have f : (A × B) → (B × A) such that f(a, b) = (b, a). (a) ∴ f{a1, b1) = (b1, a1), f(a2, b2) = (b2, a2) and f(a1, b1) = f(a2, b2) ⇒ (b1,1... 2
- Let A = R-{3} and B = R-{1}. consider the function f: A → B defined by f (x) = \(\frac{x-2}{x-3}\). f : A → B, where A = R- 3, B = R – 1, f is defined by (x1 – 2)(x2 – 3) = (x2 – 2)(x1 – 3) or x1x2 – 3x1 – 2x2 + 6 = x1x2 – 2x1 – 3x2 + 6 i.e... 1
- Let f: R → R be defined as f(x) = x4. Choose the correct answer: (a) f is one-one onto. (b) f is many-one onto. (D) f is neither one-one nor onto https://www.youtube.com/watch?v=RSbkdpekHC0 1
- Let f: R → R be defined as f(x) = 3x. Choose the correct answer: (a) f is one-one onto. (b) f is many-one onto. (a) f is one-one onto. f : R → R is defined by f(x) = 3x. (a) f(x1) = 3x1, f(x2) = 3x2 f(x1) = f(x2) ⇒ f is one-one. (b) For every member y... 1
- Determine whether each of the following relations are reflexive, symmetric and transitive: (i) Relation R in the set A = {1, 2, ….. , 14) defined as R = {(x, y) : 3x – y = 0} (a) Put y = x, 3x – x ≠ 0 ⇒ R is not reflexive. (b) If 3... 2
- Show that the relation R in the set R of real numbers, defined as R = {(a, b) : a ≤ b2} (i) R is not reflexive ∵ a is not less than or equal to a² for all a ∈ R, e.g., 1/2 is not less than 1/4 . (ii) R is not symmetric, since if... 1
- Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b): b = a + 1} (i) R is not reflexive : a ≠ a + 1. (ii) R is not symmetric : If b = a + 1, then a ≠ b + 1. (iii) R is not transitive :If b = a + 1, c = b +... 1
- Show that the relation R in R defined as R = {(a, b): a ≤ b} is reflexive and transitive but not symmetric. R = {(a, b): a ≤ b} (i) R is reflexive. Replacing b by a, a ≤ a ⇒ a = a is true. (ii) R is not symmetric : a < b and b < a which is not true... 1
- Check whether the relation R, defined by R = {(a, b) : a ≤ b3} is reflexive, symmetric or transitive. (i) a ≤ a³ is not true. e.g. 1/2 is not less than ( 1/2 )³ ∴ R is not reflexive. (ii) If a ≤ b³, then b is not less than or equal a3, e.g. 1... 1
- Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} (i) (1, 1), (2, 2), (3, 3) do not belong to relation R. ∴ R is not reflexive. (ii) It is symmetric because (1, 2) and (2, 1) belong to R. (i... 2
- Show that the relation R in the set A of all books in a library of a college, given by R = {(x, y): x (i) The number of pages in a book remain the same. ⇒ Relation R is reflexive. (ii) The book x has the same number of pages as the book y. ⇒... 1