Rational Function

6. Rational function. Definition: given polynomials $p(x),q(x)$, $f(x)=\dfrac{p(x)}{q(x)}$ is defined for $x$ whenever $q(x)\ne0$. Example: $f(x)=\dfrac1x$, $x\ne0$ (Fig. 6.27). Domain: $R-\{0\}$; Range: $R-\{0\}$.

Properties: (1) As $x\to0$ (i.e. as $x$ approaches 0), $f(x)\to\infty$ or $f(x)\to-\infty$, so the line $x=0$ (the $Y$-axis) is called a vertical asymptote. (A straight line that does not intersect the curve, but such that as $x$ approaches $\infty$ or $-\infty$ the distance between the line and the curve tends to 0, is called an asymptote of the curve.) (2) As $x\to\infty$ or $x\to-\infty$, $f(x)\to0$, so the line $y=0$ (the $X$-axis) is called a horizontal asymptote. (3) The domain of a rational function $f(x)=\dfrac{p(x)}{q(x)}$ is all real values except the zeroes of $q(x)$.

Ex. 8: Find the domain and range of the function $f(x)=\dfrac{6-4x^2}{4x+5}$.

Solution: $f(x)$ is defined for all $x\in R$ except when the denominator is 0. Since $4x+5=0\implies x=-\dfrac54$, the domain of $f(x)$ is $R-\left\{-\dfrac54\right\}$. …