f(x)=x-7/x^2-4x-12

Question

Find the domain of the rational

Answer 1

To find domain mean to find critical points where given function might not exist.
In our case we have to exclude the points where trinomial equal zero.
                x - 7         
 f(x) = --------------- 
           x² - 4x - 12 

x² - 4x - 12 = (x - 6)(x + 2) ≠ 0

x ≠ 6
x ≠ - 2

Thus, domain is (- ∞, - 2) U (-2, 6) U (6, + ∞) , in other words: all real number except "- 2" and 6 .

Answer 2

The domain means the values of x that are allowed. In the case of rational expressions, that is all values that don't result in division by zero.

 

     x - 7

-------------

x² - 4x - 12

 

         x - 7

= ----------------

   (x - 6) (x + 2)

 

 

The domain is all values of x except for -2 and 6.

Answer 3

I assume you mean (x-7) / (x^2 - 4x - 12), right?

 

If so the first step is to factor the quadratic expression in the denominator.  It factors into (x-6)(x+2).

 

Since there are no (binomial) factors in common between the numerator and the denominator, this expression can not be further simplified.

 

The domain for a polynomial expression, like the this one, is the set of all real numbers except for those values that would make the denominator equal to zero.  In this specific case, if x = 6 or x = -2, the denominator would equal zero.

 

So, the domain of this function is (- infinity, -2) U (-2, 6) U (6, positive infinity).