how to find domain and answers of a function

Question

g(x) = (64 − x^2)/(x^2 + 8x)

(a) Find the domain of g(x).

x ≠ 0

x ≠ 64, x ≠ 0

x ≠ −8

x ≠ −8, x ≠ 0

x ≠ 64

x ≠ −8, x ≠ 8

I said it was the third choice because that would make the denominator zero but I'm not sure if that's the right one.

Part B:

g(x)=0

g(x)=1

g(x)=-1

How many solutions does each equation have? my options are none, 1, 2, more than 2. For the first one I said 1, and for the second and third I said 2 but I never really understood these problems. Could you please explain them to me.

Answer 1

First you need to factor the function.

 

g(x) = [(8 - x)(8 + x)] / [x(x + 8)]

 

 

Next, you can cancel out (x+8) since it appears in the denominator and numerator.  This means at x=-8 is the location of discontinuity.

 

Now you are left with

 

g(x) = (8 - x) / x

 

 

Set the denominator equal to zero to find your restrictions.  When you know your restrictions, then you found your domain.