PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions for Mathematics

Consider f : {1, 2, 3} → {a, b, c} and g : {a, b, c} → {apple, ball, cat} defined as f (1) = a, f (2) = b, f (3) = c, g (a) = apple, g (b) = ball and g (c) =  cat. Show that f, g and gof are invertible. Find f⁻¹, g⁻¹ and gof⁻¹and show that (gof)⁻¹ = f ⁻¹o g⁻¹

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⁻¹ og⁻¹ as the sets of ordered pairs and verify that (gof)⁻¹ = f⁻¹ og⁻¹.

Show that the function f : Q → Q, defined by f(x) = 3x + 5, is invertible. Also, find f⁻¹

Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.

Consider f : R → R₊ → [4, ∞) given by f(x) = x² + 4. Show that f is invertible with inverse f⁻¹ of f given by f⁻¹ `(x)= sqrt (x-4)` where R⁺ is the set of all non-negative real numbers.

Consider f : R₊ → [−5, ∞) given by f(x) = 9x² + 6x − 5. Show that f is invertible with `f^-1 (x) = (sqrt (x +6)-1)/3 .`

If f : R → R be defined by f(x) = x³ −3, then prove that f⁻¹ exists and find a formula for f⁻¹. Hence, find f⁻¹(24) and f⁻¹ (5).

A function f : R → R is defined as f(x) = x³ + 4. Is it a bijection or not? In case it is a bijection, find f⁻¹ (3).

If f : Q → Q, g : Q → Q are two functions defined by f(x) = 2 x and g(x) = x + 2, show that f and g are bijective maps. Verify that (gof)⁻¹ = f⁻¹ og ⁻¹.

Let A = R - {3} and B = R - {1}. Consider the function f : A → B defined by f(x) = `(x-2)/(x-3).`Show that f is one-one and onto and hence find f^-1.

                    [CBSE 2012, 2014]

Consider the function f : R⁺ →  [-9 , ∞ ]given by f(x) = 5x² + 6x - 9. Prove that f is invertible with f ^-1 (y) = `(sqrt(54 + 5y) -3)/5`             [CBSE 2015]

Let f : R `{- 4/3} `- 43 →">→ R be a function defined as f(x) = `(4x)/(3x +4)` . Show that f : R - `{-4/3}`→ Rang (f) is one-one and onto. Hence, find f ^-1.

If f : R → (−1, 1) defined by `f (x) = (10^x- 10^-x)/(10^x + 10 ^-x)` is invertible, find f⁻¹.

If f : R → (0, 2) defined by `f (x) =(e^x - e^(x))/(e^x +e^(-x))+1`is invertible , find f^-1.

Let f : [−1, ∞) → [−1, ∞) be given by f(x) = (x + 1)² − 1, x ≥ −1. Show that f is invertible. Also, find the set S = {x : f(x) = f⁻¹ (x)}.

Let A = {x &epsis; R | −1 ≤ x ≤ 1} and let f : A → A, g : A → A be two functions defined by f(x) = x² and g(x) = sin (π x/2). Show that g⁻¹ exists but f⁻¹ does not exist. Also, find g−1.

Let f be a function from R to R, such that f(x) = cos (x + 2). Is f invertible? Justify your answer.

If A = {1, 2, 3, 4} and B = {a, b, c, d}, define any four bijections from A to B. Also give their inverse functions.

Let A and B be two sets, each with a finite number of elements. Assume that there is an injective map from A to B and that there is an injective map from B to A. Prove that there is a bijection from A to B.

If f : A → A, g : A → A are two bijections, then prove that fog is an injection ?