rx x2 2x 3 State the Domain and the Range in Interval Notat

r(x) =

x² 2x 3

State the Domain and the Range in Interval Notation

Solution

r(x) = 4x^2 /( x^2 -2x -3)

Domain: all real values of x for which function exists:

Points where denominator is zero, function does not exist.

So, x^2 -2x -3 =0 -----> x^2 +3x -x -3 =0

x(x+3) -1(x+3) =0 -----> (x-1)(x+3) =

x=1 and x=3

Domain: ( - infinity , 1) U ( 1 , 3) U ( 3, infinity)

Range : Find inverse of 4x^2 /( x^2 -2x -3)

Inverse : ( x- 2 sqrt(x-3) )/(x-4) ;   ( x + 2 sqrt(x-3) )/(x-4)

Find domain of inverse functions: Domain of ( x- 2 sqrt(x-3) )/(x-4) ---->( -inf, 0) or [3, 4) U ( 4, infinity)

  Domain of ( x+ 2 sqrt(x-3) )/(x-4) ---->( -inf, 0) or [3, 4) U ( 4, infinity)

So, Range of original function is same as domain of inverse function:

Range : ( -inf, 0) or [3, 4) U ( 4, infinity)