Question 77043
The given expression is: 


f(x) = {{{ (x - 2)/ (4x^2 - 5x - 6) }}} 


To find the domain of the given function, we first factorize the denominator. 


Here the denominator is {{{ 4x^2 - 5x - 6 }}} 


We find the roots of the above equation, either by factorization or by using the quadratic formula. The above expression cannot be solved by the factoring. so we use the quadratic formula, which is given by: 


{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 


{{{x = (5 +- sqrt( 5^2-4*(4)*(-6) ))/(2*4) }}} 


{{{x = (5 +- sqrt(25 + 96))/8 }}} 


{{{x = (5 +- sqrt(121))/8}}} 


{{{ x = (5 +- 11)/8 }}}


{{{ x = 16/8 }}} or {{{x = -6/8}}} 


x = 2 or {{{x = - 3/4}}} 


These are the 2 numbers which are to be excluded from the set of domain. These numbers show that when plugged into the denomiantor gives us a zero. That is these two numbers are the roots of the polynomial in the denomiantor. Hence, they must be excluded from the domain. 


Hence, the solution. 



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