Question

How do you find the domain and range of a rational function?

Answer 1

Hint: The domain of a rational function is such that, for any input value, the denominator must not be zero.

Complete step-by-step solution:
We know that a rational function, $f(x)$, is a function of $x$, which can be expressed in the form, $\dfrac{{g(x)}}{{q(x)}}$, where $g(x),\,q(x)$ are two polynomials and $q(x) \ne 0$ for any $x$.
Also, the domain of a rational function is the set of all values for which the function is defined. Now, the range of the function is the values of $f(x)$ for given values of $x$.
Thus, the domain of the function $f(x)$ is the set of values of $x$ such that $q(x) \ne 0$ while the range is the set of values of $f(x)$ corresponding to the domain.

Additional information:
A function, $f(x) = a$, where $a$ is a constant is a rational function, even though the value of $f(x)$ can be rational or irrational.
For example, $h(x) = \pi $, is a rational function even though the value of $h(x)$ for any values of $x$ remains $\pi $, which is an irrational number.

Note:
Every polynomial function is a rational function while a rational function, $f(x) = \dfrac{{g(x)}}{{q(x)}}$, is a polynomial function if $q(x) = 1$. That is the denominator is a constant polynomial function.