State with reasons whether the following function has inverse : (i) f : [1, 2, 3, 4] → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)} Watch Solution
State with reasons whether the following function has inverse : (ii) g : {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)} Watch Solution
State with reasons whether the following function has inverse : (iii) h : {2, 3, 4, 5} → {7, 9, 11, 13} with h = {(2, 7), (3, 9), (4, 11), (5, 13)} Watch Solution
Find f^{-1} if it exists for f: A → B where A = {0, -1, -3, 2}; B = {-9, -3, 0, 6} and f(x) = 3x Watch Solution
Find f^{-1} if it exists for f: A → B where A = {1, 3, 5, 7, 9}; B = {0, 1, 9, 25, 49, 81} and f(x) =x^2 Watch Solution
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}, gof^{-1} and show that (gof)^{-1} = f^{-1}og^{-1}. Watch Solution
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) = x^2 – 2. Express (gof)^{-1} and f^{-1}og^{-1} as the sets of ordered pairs and verify (gof)^{-1} = f^{-1}og^{-1}. Watch Solution
Show that the function f : Q → Q defined by f(x) = 3x + 5 is invertible. Also, find f^{-1}. Watch Solution
Show that the function f : R → R defined by f(x) = 4x + 3 is invertible. Find the inverse of f. Watch Solution
Consider f:R+→[4,∞) given by f(x)=x^2+4. Show that f is invertible with f^{-1} off given by f^−1(x)=√(x – 4) ,where R+ is the set of all non-negative real numbers. Watch Solution
If f(x) = (4x+3)/(6x-4), x≠ 2/3 show that fof(x) = x for all x≠ 2/3 . What is the inverse of f ? Watch Solution
Consider f:R+→[−5,∞) given by f(x)=9x^2 +6x-5. Show that f is invertible with f^−1(x)= {√(x+6) – 1}/3 Watch Solution
If f:R→R be defined by f(x)=x^3-3, then prove that f^-1 exists and find a formula for f^-1.Hence, find f^-1(24) and f^-1(5) Watch Solution
A function f : R → R is defined as f(x) = x^3 + 4. Is it a bijection or not? In case it is a bijection, find f^-1(3). Watch Solution
If f:Q→Q, g:Q→Q are two functions defined by f(x) =2x and g(x) =x + 2, show that f and g are bijective maps.Verify that (gof)^-1=f^-1og^-1 Watch Solution
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 Watch Solution
Consider the function f:R+→[−9,∞) given by f(x) = 5x^2+6x−9. Prove that f is invertible with f^-1(y)={√(54+5y)-3}/5 Watch Solution
Let f:N→N be a function defined as f(x)=9x^2+6x-5. Show that f:N→S, where S is the range of f is invertible. Find the inverse of f and hence find f^-1(43) and f^-1(163). Watch Solution
Let f:R-{-4/3}→R be a function defined as f(x)=4x/(3x+4). Show that f:R-{-4/3}→ range (f) is one-one and onto. Hence, find f^-1. Watch Solution
If f:R→(-1,1) defined by f(x)=(10^x-10^-x)/(10^x+10^-x) is invertible, find f^-1. Watch Solution
If f:R→(0,2) defined by f(x) = (e^x-e^-x)/(e^x+e^-x) is invertible, find f^-1. Watch Solution
Let f : [-1, ∞)→[-1, ∞) is given by f(x) = (x+1)^2 – 1. Show that f is invertible. Also, find the set S = {x: f(x) = f^-1(x)} Watch Solution
Let A ={x ϵ R | -1 ≤ x ≤ 1} and let f:A → A, g :A→A be two functions defined by f(x) = x^2 and g(x) = sin πx/2. Show that g^-1 exists but f^-1 does not exist. Also, find g^-1. Watch Solution
Let f be a function from R to R such that f(x) = cos (x + 2). Is f invertible? Justify your answer. Watch Solution
If A = {1, 2, 3, 4} and B = {a, b, c, d}, define any four bijections from A to B. Also, give their inverse function. Watch Solution
Let A and B be two sets each with 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. Watch Solution
If f : A → A and g : A → A are two bijections, then prove that (i) fog is an injection (ii) fog is a surjection Watch Solution
