find all values of x that are not in the domain of h

Question

$$ h(x) = \frac{x-7}{x^2 - 3x -10} $$

Can someone help, I need to find all values of x that are not in the domain of h

Answer 1

The fact that this looks like a homework problem aside, consider what it means for a value of $x$ to not be in the domain of $h$. It means that $h(x)$ does not return a value. So where might this happen? Well the numerator will return a value for every $x$. So that means something has to be happening in the denominator. Recall that dividing by $0$ is impossible. So now your task becomes finding for which $x$ the denominator is equal to $0$.

Answer 2

Hint :

Write it as $$f\left( x \right) =\frac { x-7 }{ { x }^{ 2 }-3x-10 } =\frac { x-7 }{ \left( x+2 \right) \left( x-5 \right) } \\ $$

Answer 3

Let $f(x)=\frac{x-7}{x^2 -3x -10}$. To find the domain of any rational function (recognize that $f(x)$ is a rational function) we exclude those values of $x$ at which the denominator is equal to zero.

So for what values of $x$ is the denominator $x^2-3x-10$ equal to zero? This is the same as finding the values of $x$ such that $x^2-3x-10=0$. You can find the roots via the quadratic formula and factoring, or just factoring via guess and check.

Either way you will find that $f(x)=x^2-3x-10=(x+2)(x-5)$. Thus the values of $x$ at which the denominator will evaluate to zero $x=-2$ and $x=5$.

These are the two values of $x$ which we will want to exclude from the domain.

In conclusion, the domain of $f(x)$ is $\{ x \in \mathbb{R}: x \neq -2, x \neq 5 \}$