what is the domain of the function x^2+6x-16 over x^2-2x-15?

To find the domain of a rational function, we need to consider the values of x for which the function is defined. In this case, the function is defined for all values of x except those that make the denominator equal to zero.

To find when the denominator, x^2 - 2x - 15, is equal to zero, we can set the denominator equal to zero and solve for x:

x^2 - 2x - 15 = 0

To factorize the quadratic equation, we look for values of a and b such that a + b = -2 and a * b = -15. In this case, a = -5 and b = 3 satisfy these conditions:

(x - 5)(x + 3) = 0

Setting each factor equal to zero:

x - 5 = 0 or x + 3 = 0

Solving for x:

x = 5 or x = -3

Therefore, the values x = 5 and x = -3 make the denominator zero, so they are not part of the domain of the function.

Therefore, the domain of the function f(x) = (x^2 + 6x - 16)/(x^2 - 2x - 15) is all real numbers except x = 5 and x = -3.