Question

Set theory

Find f(A) and f-(B) for the given function and sets. (a) f : R → R is defined by f(x) = x2 + 1, A = [-1,2], B = [0,4]. (b) f R R is defined by f(x) - sin z, A [0, ], B [0,2]. (c) f : R → Z is defined by f(x)-번 (the floor function), A = (0,4], B-(0,1,2). (d) f : R-(0) → R is defined by f(z) = x + 1, As [1,5], B = [0,1).

Answer 1

f(A) and f^-1 (B) are the function a) f: R rightarrow R and f(x) = x^2 + 1 A = [- 1, 2] B = [0, 4] f(A) = {f(x) elementof R, x elementof A} as - 1 elementof A then f(- 1) = (- 1)^2 + 1 = 2 f(0) = 0^2 + 1 = 1 f(2) = 2^2 + 1 = 5 f(- 1/2) = 1/4 + 1 = 5/4 f(1/2) = 1/4 + 1 = 5/4 so x elementof A, f(x) elementof [1, 5] f(A) = {[1, 5]} f(x) = x^2 + 1 let I = x^2 + 1 then x^2 = y - 1 x = plusminus y - 1 let I elementof B = [0, 4] then y = 0 rightarrow x is not defined in R b) Given f(x) = sin x A = [0, 11] , B = [0, 2] x elementof A if let x = 0 then f(0) = 0 x = pi/4 then f(pi/4) = 1/squareroot 2 x = pi/2 f(pi/2) = 1 x = 3 pi/2 f(3 pi/2) = 0 f(A) = { f(x) >