Show that the function f:R to R defined by f(x) = x /x2 + 1 for all x belongs to r is neither one-one nor onto. Also if g: R to R is defined as g(x) = 2x – 1 . find f o g(x)

Asked by haroonrashidgkp | 6th Jun, 2018, 12:25: PM

Expert Answer:

begin mathsize 16px style straight f left parenthesis straight x right parenthesis equals fraction numerator straight x over denominator straight x squared plus 1 end fraction
for space all space straight x greater than 0.............. space straight f left parenthesis straight x right parenthesis element of left parenthesis 0 comma 1 right parenthesis
for space straight x equals 0 comma space straight f left parenthesis straight x right parenthesis equals 0
for space all space straight x less than 0.............. space straight f left parenthesis straight x right parenthesis element of left parenthesis negative 1 comma 0 right parenthesis
range space open parentheses negative 1 comma 1 close parentheses comma space so space not space onto space since space domain space is space open parentheses negative straight R comma straight R close parentheses
differenciate
straight f apostrophe left parenthesis straight x right parenthesis equals fraction numerator 1 minus straight x squared over denominator open parentheses straight x squared plus 1 close parentheses squared end fraction
straight f apostrophe left parenthesis straight x right parenthesis equals 0 space at space straight x equals 1
so space slope space changes space after space straight x equals 1
so space there space exists
straight x subscript 1 space and space straight x subscript 2 space for space which space straight f left parenthesis straight x subscript 1 right parenthesis equals straight f left parenthesis straight x subscript 2 right parenthesis
straight f left parenthesis straight x right parenthesis space is space not space one minus one

straight g left parenthesis straight x right parenthesis equals 2 straight x minus 1
straight f left parenthesis straight g left parenthesis straight x right parenthesis right parenthesis equals fraction numerator 2 straight x minus 1 over denominator open parentheses 2 straight x minus 1 close parentheses squared plus 1 end fraction
end style

Answered by Arun | 7th Jun, 2018, 09:53: AM

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