SOLUTION: Which of the following functions have inverses: a) f(x)=absolute value of x-6 b) f(x)=ax + b, a not equal to 0 c) f(x)=x^3-19

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Question 33774: Which of the following functions have inverses:
a) f(x)=absolute value of x-6
b) f(x)=ax + b, a not equal to 0
c) f(x)=x^3-19
Answer by venugopalramana(3286) About Me  (Show Source):
You can put this solution on YOUR website!
Which of the following functions have inverses:LET Y=F(X)
THIS MEANS FOR A GIVEN VALUE OF Y THERE SHOULD BE A UNIQUE X..OR.IF Y1=Y2 THEN X1=X2
a) f(x)=absolute value of x-6....IT HAS INVERSE IF WE DEFINE IT AS DUAL FUNCTION FOR DIFFERENT VALUES OF X.SINCE Y=|X-6|...OR..Y=X-6...IF X>6....THEN X=Y+6......BUT....Y=-(X-6)....IF X<6....THEN....X=-Y+6....IT SATISFIES THE ABOVE CONDITION..IF WE ADOPT THE FUNCTION DIFFERENTLY FOR DIFFERENT RANGES OF X..OTHERWISE,IT WILL NOT BE A FUNCTION OR WILL NOT HAVE AN INVERSE..SINCE FOR Y=3 X COULD BE -3...OR....3
b) f(x)=ax + b, a not equal to 0.....IT HAS INVERSE..SINCE X=(Y-B)/A..WITH A NOT EQUAL TO ZERO.
c) f(x)=x^3-19....IT HAS INVERSE IN REAL FIELD,BUT NOT IN COMPLEX FIELD..SINCE X=CUBE ROOT OF (Y+19) HAS UNIQUE ROOT IN REAL FIELD BUT 2 IMAGINARY AND 1 REAL ROOT IN COMPLEX FIELD.