F:z->z given by f(x)=x2+4 is this function  one one and onto ?

Asked by Kalyanrao Chavan | 21st Aug, 2014, 07:33: AM

Expert Answer:

Here space we space have space to space show space that space straight f left parenthesis straight x right parenthesis space is space both space one minus one space and space onto. For space one minus one comma straight f left parenthesis straight x right parenthesis equals straight x squared plus 4 comma space straight f left parenthesis straight y right parenthesis equals straight y squared plus 4 Now comma space for space straight f left parenthesis straight x right parenthesis space equals space straight f left parenthesis straight y right parenthesis rightwards double arrow straight x squared plus 4 equals straight y squared plus 4 rightwards double arrow straight x squared equals straight y squared rightwards double arrow straight x equals straight y because for space straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight y right parenthesis rightwards double arrow straight x equals straight y comma space therefore space straight f left parenthesis straight x right parenthesis space is space straight a space one minus one left parenthesis injective right parenthesis space function For space onto comma straight f left parenthesis straight x right parenthesis equals straight y rightwards double arrow straight x squared plus 4 equals straight y rightwards double arrow straight x squared equals straight y minus 4 rightwards double arrow straight x equals square root of straight y minus 4 end root Clearly space square root of straight y minus 4 end root space is space straight a space real space number space for space all space straight y greater than 4. Thus space for space all space straight y element of straight R comma space there space exists space straight x equals square root of straight y minus 4 end root space element of straight R comma space such space that straight f left parenthesis straight x right parenthesis equals straight f open parentheses square root of straight y minus 4 end root close parentheses equals open parentheses square root of straight y minus 4 end root close parentheses squared plus 4 equals straight y minus 4 plus 4 equals straight y This space shows space that space every space element space in space co minus domain space has space its space preimage space in space domain. Hence space straight f left parenthesis straight x right parenthesis space is space onto left parenthesis surjective right parenthesis. Thus space straight f left parenthesis straight x right parenthesis space is space both space one minus one space and space onto.

Answered by Prasenjit Paul | 21st Aug, 2014, 10:53: AM