what is the domain of the function? \n$f(x)=\\frac{x + 6}{(x - 7)(x + 5)}$\na all real numbers except 7\nb…

what is the domain of the function? \n$f(x)=\\frac{x + 6}{(x - 7)(x + 5)}$\na all real numbers except 7\nb all real numbers except 5 and -7\nc all real numbers except -5 and 7\nd all real numbers

what is the domain of the function? \n$f(x)=\\frac{x + 6}{(x - 7)(x + 5)}$\na all real numbers except 7\nb all real numbers except 5 and -7\nc all real numbers except -5 and 7\nd all real numbers

Answer

Explanation:

Step1: Recall domain of rational function

For a rational function ( f(x)=\frac{g(x)}{h(x)} ), the domain excludes values where ( h(x) = 0 ) (since division by zero is undefined). Here, ( h(x)=(x - 7)(x + 5) ).

Step2: Solve ( h(x)=0 )

Set ( (x - 7)(x + 5)=0 ). By zero - product property, if ( ab = 0 ), then ( a = 0 ) or ( b = 0 ). So, ( x-7 = 0 ) gives ( x = 7 ), and ( x + 5=0 ) gives ( x=-5 ). Thus, the values ( x = 7 ) and ( x=-5 ) make the denominator zero, so they are excluded from the domain.

Answer:

C. all real numbers except (-5) and (7)