Characteristics of rational functions

Domain

  • It must be determined before simplifying the function.
  • The only restriction on the domain of a rational function is division by zero.
  • Zeros (\(x\)-intercepts) of the denominator are disallowed in the domain of a rational function.
  • The denominator of a rational function is a polynomial, which does not always have zeros. Therefore, not every rational function has restrictions on its domain.
Examples
  • The domain of \(y=\frac{1}{x^2+1}\) is all real numbers, because \(x^2+1=0\) has no solutions.
  • The domain of \(y=\frac{1}{x^2-1}\) is \(\{x\in\mathbb{R}\mid x\neq\pm 1\}\) because the solution of \(x^2-1=0\) is \(x=\pm1\).

Range

  • There is no solid single rule for determining the range, and every case has to be considered individually.
Examples
  • The function \(y=\frac{1}{x^2+1}\) has the range \(\{y\in\mathbb{R}\mid 0\lt y\le 1\}\).
  • The range of \(y=\frac{1}{x^2}\) is \(\{y\in\mathbb{R}\mid y\gt 0\}\).
  • The range of \(y=\frac{1}{x}\) is \(\{y\in\mathbb{R}\mid y\neq 0\}\).

\(\boldsymbol{x}\)-intercepts

  • A rational function has \(x\)-intercepts wherever its numerator is zero,
  • provided that the function is defined at that value of \(x\).
  • To get the \(x\)-intercepts, we set the numerator of the simplified function equal to zero, and solve the equation.
  • The numerator is a polynomial, which may or may not have zeros. Therefore, a rational function may or may not have zeros.
Examples
  • The function \(y=\frac{x^2+1}{x-2}\) does not have zeros, because \(x^2+1=0\) has no zeors.
  • The function \(y=\frac{x^2-1}{x-2}\) has zeros at \(x=\pm1\), because these are the zeros of \(x^2-1=0\).

\(\boldsymbol{y}\)-intercept

  • A function has a \(y\)-intercept wherever \(x=0\).
  • Therefore, a rational function has a \(y\)-intercept if and only if it is defined at \(x=0\).
Examples
  • The function \(y=\frac{1}{x}\) does not have \(y\)-intercept because it is undefined at \(x=0\).
  • The function \(y=\frac{1}{x-1}\) has a \(y\)-intercept at \((0,-1)\).

Vertical asymptotes

A rational function has vertical asymptotes wherever the simplified denominator is zero. Because the denominator is a polynomial, a rational function may or may not have vertical asymptotes.

A function never crosses its vertical asymptote.

Horizontal or oblique asymptote

These depend on the degrees of both the numerator and denominator. Let \(\operatorname{deg}(P)\) be the degree of the numerator and \(\operatorname{deg}(Q)\) be that of the denominator.

  • If \(\operatorname{deg}(P)\lt\operatorname{deg}(Q)\), the function has a horizontal asymptote at \(y=0\)
  • If \(\operatorname{deg}(P)=\operatorname{deg}(Q)\), the function has a horizontal asymptote at \(y=\dfrac{\text{Leading coefficient of numerator}}{\text{Leading coefficient of denominator}}\)
  • If \(\operatorname{deg}(P)=\operatorname{deg}(Q)+1\), the function has an oblique asymptote at \(y=\text{quotient of }\dfrac{P(x)}{Q(x)}\)
  • If \(\operatorname{deg}(P)\gt\operatorname{deg}(Q)+1\), the function has neither a horizontal nor an oblique asymptote

A rational function may cross its horizontal or oblique asymptote, but beyond a certain value of \(x\) (depending on the function) it will just keep approaching without touching it.

End behaviour

If a rational function has a horizontal asymptote, then \(y\) approaches the asymptote as \(x\) approaches \(\pm\infty\). To check whether it approaches from above or from below, evaluate the function for a relatively high value of \(x\) (e.g., \(x=100\) or \(x=1000\)) and compare that to the location of the asymptote.

If a rational function has an oblique asymptote or neither horizontal nor oblique asymptotes, then \(y\) approaches \(\pm\infty\) as \(x\) approaches \(\pm\infty\). To check which infinity it approaches, evaluate the function at a relatively high value of \(x\).

Symmetry

Depending on its exact definition, a rational function may be even, odd or neither. We can check this algebraically as follows.

  • A function is odd if and only if \(f(-x)=-f(x)\).
  • A function is even if and only if \(f(-x)=f(x)\).
  • Otherwise, the function is neither odd nor neve.

Behaviour of a rational function

  1. Factor the numerator and the denominator. DO NOT simplify yet.
  2. Factors in the denominator give us restrictions on the domain. Denominator cannot be zero.
  3. Simplify
  4. Factors that disappear from the denominator give us holes. To know the \(y\)-coordinate of the hole, substitute with the \(x\)-value into the simplified function.
  5. Factors that remain in the denominator give us vertical asymptotes.
  6. Factors that remain in the numerator give us \(\boldsymbol x\)-intercepts provided that the function is defined there (i.e., does not have holes or vertical asymptotes).
  7. To get the \(y\)-intercept, substitute with \(x=0\) (if it the function is defined there).
  8. To determine whether the function has horizontal or oblique asymptotes, compare the degrees of the numerator and denominator as described above.