Question 97438
{{{f(x)= (x^4-2x^3+7) / (3x^2-10x-8)}}} Start with the given function



{{{3x^2-10x-8=0}}} Set the denominator equal to zero


{{{(x-4)(3x+2)=0}}} Factor the left side (note: if you need help with factoring, check out this <a href=http://www.algebra.com/algebra/homework/playground/change-this-name4450.solver>solver</a>)




Now set each factor equal to zero:


{{{x-4=0}}} or {{{3x+2=0}}}


{{{x=4}}} or {{{x=-2/3}}}  Now solve for x in each case



Since {{{x=4}}} or {{{x=-2/3}}} make the denominator equal to zero, that means we must exclude these values from the domain.


So our domain is: x is the set of all real numbers except {{{x<>4}}} or {{{x<>-2/3}}}



Which looks like this in interval notation:

*[Tex \Large \left(-\infty, -\frac{2}{3}\right)\cup\left(-\frac{2}{3}, 4\right)\cup\left(4,\infty \right)]