a)If g(x) = x^2+x-2 and 1/2gof (x) = 2x^2-5x+2 then find f(x)
 
b)if f(x) = sin^2 x and g(f(x)) = l sin x l  then find g(x)
 
note : x^2 means x square
 
sin^2 x means sin square x

Asked by ahuja8087 | 24th Jun, 2016, 12:59: AM

Expert Answer:

begin mathsize 12px style left parenthesis straight a right parenthesis space straight g left parenthesis straight x right parenthesis equals straight x squared plus straight x minus 2
1 half straight g left parenthesis straight f left parenthesis straight x right parenthesis right parenthesis equals 2 straight x squared minus 5 straight x plus 2
rightwards double arrow straight g left parenthesis straight f left parenthesis straight x right parenthesis right parenthesis equals 4 straight x squared minus 10 straight x plus 4... left parenthesis straight i right parenthesis

Here space straight g left parenthesis straight f left parenthesis straight x right parenthesis right parenthesis space is space straight a space quadratic space expression. space So space is space straight g left parenthesis straight x right parenthesis.
So space straight f left parenthesis straight x right parenthesis space has space to space be space straight a space linear space expression.
Suppose space straight f left parenthesis straight x right parenthesis space equals space ax plus straight b
straight g left parenthesis straight f left parenthesis straight x right parenthesis right parenthesis equals open parentheses ax plus straight b close parentheses squared plus open parentheses ax plus straight b close parentheses minus 2
Simplifying space and space comparing space with space left parenthesis straight i right parenthesis comma space we space get
straight a squared equals 4 comma space 2 ab plus straight a equals negative 10 space and space straight b squared plus straight b minus 2 equals 4
Solving space first space two space equations space and space verifying space in space third comma space we space get
two space solutions colon
straight a equals negative 2 comma space straight b equals 2
and
straight a equals 2 comma space straight b equals negative 3
Hence comma space straight f left parenthesis straight x right parenthesis equals negative 2 straight x plus 2 space
or space straight f left parenthesis straight x right parenthesis equals 2 straight x minus 3

Kindly space ask space one space question space in space one space post. end style

Answered by satyajit samal | 24th Jun, 2016, 12:54: PM