Question 1:Let f : {1, 3, 4} → {1, 2, 5} and g : {1, 2, 5} → {1, 3} be given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down gof.
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Question 2:Let f, g and h be functions from R to R. Show that (f + g)oh = foh + goh (f . g)oh = (foh) . (goh)
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Question 3:Find gof and fog, if (i) f (x) = | x | and g(x) = | 5x – 2 | (ii) f (x) = 8x3 and g(x) = 1 x3 .
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Question 4:If f (x) = (4 3) (6 4) x x + − , 2 3 x ¹ , show that fof (x) = x, for all 2 3 x¹ . What is the inverse of f ?
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Question 5:State with reason whether following functions have inverse (i) f : {1, 2, 3, 4} → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)} (ii) g : {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)} (iii) h : {2, 3, 4, 5} → {7, 9, 11, 13} with h = {(2, 7), (3, 9), (4, 11), (5, 13)}
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Question 6:Show that f : [–1, 1] → R, given by f (x) = ( 2) x x + is one-one. Find the inverse of the function f : [–1, 1] → Range f. (Hint: For y Î Range f, y = f (x) = 2 x x + , for some x in [–1, 1], i.e., x = 2 (1 ) y − y )
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Question 7:Consider f : R → R given by f (x) = 4x + 3. Show that f is invertible. Find the inverse of f.
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Question 8:Consider f : R + → [4, ∞) given by f (x) = x2 + 4. Show that f is invertible with the inverse f –1 of f given by f –1(y) = y − 4 , where R+ is the set of all non-negative real numbers.
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Question 9:Consider f : R+ → [– 5, ∞) given by f (x) = 9x2 + 6x – 5. Show that f is invertible with f –1(y) = ( 6 ) 1 3 y + − .
1. Let f : {1, 3, 4} {1, 2, 5} and g : {1, 2, 5}
Question 10:Let f : X → Y be an invertible function. Show that f has unique inverse. (Hint: suppose g 1 and g 2 are two inverses of f. Then for all y Î Y, fog1(y) = 1Y(y) = fog2(y). Use one-one ness of f).
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Question 11:Consider f : {1, 2, 3} → {a, b, c} given by f (1) = a, f (2) = b and f (3) = c. Find f –1 and show that (f –1)–1 = f.
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Question 12:Let f : X → Y be an invertible function. Show that the inverse of f –1 is f, i.e., (f –1)–1 = f.
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Question 13:If f : R → R be given by f (x) = 1 (3 − x3 )3 , then fof (x) is (A) 1 x3 (B) x3 (C) x (D) (3 – x3).
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Question 14:Let f : R – 4 3 − → R be a function defined as f (x) = 4 3 4 x x + . The inverse of f is the map g : Range f → R – 4 3 − given by (A) 3 ( ) 3 4 y g y y = − (B) 4 ( ) 4 3 y g y y = − (C) 4 ( ) 3 4 y g y y = − (D) 3 ( ) 4 3 y g y y
