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f / [- 2, 2] -> [- 2, 2], g / [- 2, 2] -> [0, 4] are two functions defined as f(x)= -2,-2<= x<=0\\x ^ 2 - 2,0<= x<=2 . and g(x) = |f(x)| + f(|x|) then
Asked by chinna99tv | 14 Jul, 2022, 07:08: AM
Expert Answer
straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell negative 2 comma space space space space space minus 2 less or equal than straight x less or equal than 0 end cell row cell straight x squared minus 2 comma space space space space 0 less or equal than straight x less or equal than 2 end cell end table close
rightwards double arrow straight g left parenthesis straight x right parenthesis equals straight f left parenthesis vertical line straight x vertical line right parenthesis plus vertical line straight f left parenthesis straight x right parenthesis vertical line equals open curly brackets table attributes columnalign left end attributes row cell straight x squared minus 2 plus vertical line minus 2 vertical line comma space space space space space space space space space minus 2 less or equal than straight x less or equal than 0 end cell row cell straight x squared minus 2 plus vertical line straight x squared minus 2 vertical line comma space space space space space space space space space 0 less or equal than straight x less or equal than square root of 2 end cell row cell straight x squared minus 2 plus open vertical bar straight x squared minus 2 close vertical bar comma space space space space space space space square root of 2 less or equal than straight x less or equal than 2 end cell end table close
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals open curly brackets table attributes columnalign left end attributes row cell straight x squared minus 2 plus 2 comma space space space space space space space space minus 2 less or equal than straight x less or equal than 0 end cell row cell straight x squared minus 2 minus left parenthesis straight x squared minus 2 right parenthesis comma space space space space space space space 0 less or equal than straight x less or equal than square root of 2 end cell row cell straight x squared minus 2 plus straight x squared minus 2 comma space space space space space space space square root of 2 less or equal than straight x less or equal than 2 end cell end table close
rightwards double arrow straight g left parenthesis straight x right parenthesis equals straight f left parenthesis vertical line straight x vertical line right parenthesis plus vertical line straight f left parenthesis straight x right parenthesis vertical line equals open curly brackets table attributes columnalign left end attributes row cell straight x squared comma space space space space space space space space space space space minus 2 less or equal than straight x less or equal than 0 end cell row cell 0 comma space space space space space space space space space space space space space space space space 0 less or equal than straight x less or equal than square root of 2 end cell row cell 2 straight x squared minus 4 comma space space space space square root of 2 less than straight x less or equal than 2 end cell end table close
Thus, f and g are not injective but they are surjective.
Answered by Renu Varma | 19 Jul, 2022, 11:20: AM
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