Question

Question #55b9f

Answer 1

The domain is restricted by the condition that the denominator is not equal to 0.

Explanation:

Division by 0 would cause the expression be undefined, therefore, the domain restrictions can be found by solving the equation:

`x^2 + 6x - 16 = 0 " [1]"`

and then stipulating that the x cannot equal these values.

Factor equation [1]:

`(x -2)(x + 8) = 0`

This separates into two equations:

`x -2 = 0` and `x + 8 = 0`

Simplify both:

`x = 2` and `x = -8`

Therefore, the domain restrictions are `x!=2` and `x!=-8`

Answer 2

`(-oo, -8) uu (-8,2) uu (2,oo)`

Explanation:

The denominator of the function cannot be zero. To find the restrictions on the domain, find the values of `x` which would make the denominator `0`.

`x^2 + 6x - 16 != 0`

`(x+8)(x-2) != 0` `->` factor

`x+8 != 0` and `x-2 != 0`

`x !=-8` and `x !=2`

So, the domain of the function is all real numbers, where `x != -8,2`. We can write this in interval notation as `(-oo, -8) uu (-8,2) uu (2,oo)`.

We can verify this by graphing the function. As you can see, there are vertical asymptotes at `x=-8` and `x=2`.

graph{(x^2+3)/(x^2 + 6x - 16) [-17.96, 14.08, -6.02, 10.01]}