(A) g(y) = 3y/(3-4y) (B) g(y) = 4y/(4-3y) (C) g(y) =...
Let f : R â { -4/3 } â R be a function defined as f(x) = ????????/3????+????. The inverse of f is the map g : Range f â R â { -4/3 } given by
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Exercise 1.3
Consider f : {1, 2, 3} â {a, b, c} given by f(1) = a, f(2) = b and f(3) = c. Find fâ1 and show that (fâ1)â1 = f.
solution: Think about f : {1, 2, 3} â {a, b, c} given by f(1) = a, f(2) = b and f(3) = c So f = {(a, 1), (b, 2), (c, 3)} Subsequently f-1 (a) = 1, f-1 (b) = 2 and f-1 (c) = 3 Now, f-1 = {(a, 1), (b,...
Let f : X â Y be an invertible function. Show that f has unique inverse.
(Hint: suppose g1 and g2 are two inverses of f. Then for all y â Y,fog1(y) = 1Y(y) = fog2(y). Use one-one ness of f) solution: Given, f : X â Y be an invertible capacity. What's more, g1 and g2 are...
Consider f : R+ â [â 5, â) given by f (x) = 9×2 + 6x â 5. Show that f is invertible with
solution: Think about f : R+ â [â5, â) given by f (x) = 9x2 + 6x â 5 Consider f : R+ â [4, â) given by f(x) = x2 + 4 Let x, y â R â [-5, â) then, at that point f(x) = 9x2 + 6x â 5 and f(y) = 9y2 +...
Consider f : R+ â [4, â) given by f(x) = x2 + 4. Show that f is invertible with the inverse fâ1 of f given by fâ1(y) = â???? â ???? , where R+ is the set of all non-negative real numbers.
solution: Think about f : R+ â [4, â) given by f(x) = x2 + 4 Let x, y â R â [4, â) then, at that point f(x) = x2 + 4 and f(y) = y2 + 4 on the off chance that f(x) = f(y) x2 + 4 = y2 + 4 or x = y f...
Consider f : R â R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.
solution: Think about f : R â R given by f(x) = 4x + 3 Say, x, y â R Let f(x) = f(y) then, at that point 4x + 3 = 4y + 3 x = y f is one-one capacity. Let y â Range of f y = 4x + 3 or on the other...
Show that f : [â1, 1] â R, given by f (x) = x/(x+2) is one-one. Find the inverse of the function f : [â1, 1] â Range f. some x in [â1, 1], i.e., x = 2y/(1-y).
solution: Given capacity: (x) = x/(x+2) Let x, y â [â1, 1] Let f(x) = f(y) x/(x+2) = y/(y+2) xy + 2x = xy + 2y x = y f is one-one. Once more, Since f : [â1, 1] â Range f is onto say, y = x/(x+2) yx...
State with reason whether following functions have inverse
(i) f : {1, 2, 3, 4} â {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)} (ii) g : {5, 6, 7, 8} â {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)} (iii) h : {2, 3, 4, 5} â {7, 9, 11, 13} with...
If f(x) = (????????+????)/(60-4) , x â 2/3, Show that fof(x) = x, for all x â 2/3. What is the inverse of f.
solution: f(x) = (4????+3)/(60-4), x â 2/3, (6????â4) In this way, fof(x) = x for all x â 2/3. Once more, fof = I The backwards of the given capacity, f will be f.
Find gof and fog, if
(i) f(x) = | x | and g(x) = | 5x â 2 | (ii) f(x) = 8x3 and g(x) = x1/3 . solution: (I) f(x) = | x | and g(x) = | 5x â 2 | gof =(gof)(x) = g(f(x) = g(|x|) = |5 |x| - 2| mist = (fog)(x) = f(g(x)) =...
Let f, g and h be functions from R to R. Show that (f + g) oh = foh + goh (f . g) oh = (foh) . (goh)
solution: LHS = (f + g) gracious = (f+g)(h(x)) = f(h(x)) + g(h(x)) = foh + goh = RHS Once more, LHS = (f . g) gracious = f.g(h(x)) = f(h(x)) . g(h(x)) = (foh) . (goh) =...
Let f : {1, 3, 4} â {1, 2, 5} and g : {1, 2, 5} â {1, 3} be given by and g = {(1, 3), (2, 3), (5, 1)}. Write down gof.
solution: Given capacity, f : {1, 3, 4} â {1, 2, 5} and g : {1, 2, 5} â {1, 3} be given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)} Find gof. At f(1) = 2 and g(2) = 3, gof is...