Question

classify the given function as a polynomial function, rational function, or root function, and then find the domain. write the domain in interval notation. h(x) = (x^2 + 2)/(x^2 + 3x - 4) polynomial function rational function root function the domain of h(x)=(x^2 + 2)/(x^2 + 3x - 4) is (type your answer in interval notation. simplify your answer. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall definition of rational function

A rational function is of the form $\frac{f(x)}{g(x)}$ where $f(x)$ and $g(x)$ are polynomials and $g(x)
eq0$. Here, $h(x)=\frac{x^{2}+2}{x^{2}+3x - 4}$, with $f(x)=x^{2}+2$ and $g(x)=x^{2}+3x - 4$ which are polynomials, so it's a rational function.

Step2: Find values that make denominator zero

Set $x^{2}+3x - 4 = 0$. Factor the quadratic: $(x + 4)(x - 1)=0$. Solving gives $x=-4$ or $x = 1$.

Step3: Determine the domain

The domain of a rational function is all real - numbers except the values that make the denominator zero. So the domain is $(-\infty,-4)\cup(-4,1)\cup(1,\infty)$.

Answer:

$(-\infty,-4)\cup(-4,1)\cup(1,\infty)$