Answer
a. {$x$|$x$ is a real number, $x\ne-2,\frac{3}{2}$}
b. $(-∞,-2)U(-2,\frac{3}{2})U(\frac{3}{2},∞)$
Work Step by Step
We are given the function $f(x)=\frac{3x+1}{2x^{2}+x-6}$. The domain of the function will be all values of $x$ such that the denominator does not equal 0.
Therefore, we can set the denominator equal to 0. We will exclude from the domain all values of $x$ that make the denominator equal 0.
$2x^{2}+x-6=0$
This factors into $(2x-3)(x+2)=0$.
Set both factors equal to 0.
$2x-3=0$, so $x=\frac{3}{2}$
$x+2=0$, so $x=-2$
Therefore, the domain is {$x$|$x$ is a real number, $x\ne-2,\frac{3}{2}$} in set-builder notation and $(-∞,-2)U(-2,\frac{3}{2})U(\frac{3}{2},∞)$ in interval notation.