Rational Function in Algebra
A rational function in math algebra is any function that can be written as a quotient of two polynomials, typically in the form \(f(x)=\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials and the denominator is not allowed to be zero. The main algebra skills are finding the domain and predicting the graph using intercepts, holes, and asymptotes.
1) Definition and Core Restriction
A function is a rational function if it can be expressed as \(f(x)=\frac{P(x)}{Q(x)}\) with \(Q(x)\neq 0\). The restriction \(Q(x)\neq 0\) is what creates discontinuities (holes or vertical asymptotes) and determines the domain.
Key idea: The only inputs excluded from the domain of a rational function are exactly those values of \(x\) that make the denominator \(Q(x)\) equal to zero.
2) Domain of a Rational Function
The domain is the set of all real numbers except where the denominator becomes zero.
- Set the denominator equal to zero: \(Q(x)=0\).
- Solve for the excluded values \(x=a, x=b,\dots\).
- Write the domain as “all real numbers except those values,” using intervals or set notation.
If a factor cancels between numerator and denominator, the canceled \(x\)-value is still excluded from the original domain; this produces a hole (removable discontinuity) rather than a vertical asymptote.
3) Features That Determine the Graph
| Feature | How to find it (algebra) | Meaning on the graph |
|---|---|---|
| x-intercepts | Solve \(P(x)=0\) after simplifying, and verify the denominator is not zero at those \(x\)-values. | Points where the graph crosses the x-axis. |
| y-intercept | Compute \(f(0)\), provided \(0\) is in the domain (the denominator is not zero at \(x=0\)). | Point where the graph crosses the y-axis. |
| Hole (removable discontinuity) | If \((x-a)\) cancels, then \(x=a\) is excluded. The hole’s y-value is found by plugging \(x=a\) into the simplified expression. | A missing point on an otherwise smooth branch. |
| Vertical asymptote | After canceling common factors, solve the remaining simplified denominator \(Q_s(x)=0\). | The function grows without bound near that \(x\)-value. |
| Horizontal asymptote | Compare degrees. If \(\deg P<\deg Q\), then \(y=0\). If \(\deg P=\deg Q\), then \(y=\frac{\text{leading coefficient of }P}{\text{leading coefficient of }Q}\). | End behavior as \(x\to\pm\infty\). |
| Slant (oblique) asymptote | If \(\deg P=\deg Q+1\), perform polynomial long division: \(f(x)=\text{(linear)}+\frac{\text{remainder}}{Q(x)}\). | End behavior approaches a line. |
4) Worked Example of a Rational Function
Consider the rational function \(f(x)=\frac{x^2-1}{x^2-4x+3}\). Factor both polynomials: \(x^2-1=(x-1)(x+1)\) and \(x^2-4x+3=(x-1)(x-3)\).
Domain: The original denominator is zero at \(x=1\) and \(x=3\), so the domain is all real numbers except those values: \(\mathbb{R}\setminus\{1,3\}\).
Simplify the expression by canceling the common factor (while keeping domain restrictions): \(f(x)=\frac{(x-1)(x+1)}{(x-1)(x-3)}=\frac{x+1}{x-3}\), valid for \(x\neq 1\) and \(x\neq 3\).
- Hole: The canceled factor corresponds to a hole at \(x=1\). The missing y-value is found from the simplified form: \(y=\left.\frac{x+1}{x-3}\right|_{x=1}=\frac{1+1}{1-3}=\frac{2}{-2}=-1\). The hole is \((1,-1)\).
- Vertical asymptote: The remaining denominator is zero when \(x-3=0\), so the vertical asymptote is \(x=3\).
- x-intercept: Set the simplified numerator to zero: \(x+1=0\Rightarrow x=-1\). Since \(-1\) is allowed, the x-intercept is \((-1,0)\).
- y-intercept: Evaluate at \(x=0\): \(f(0)=\frac{0+1}{0-3}=-\frac{1}{3}\). The y-intercept is \((0,-\frac{1}{3})\).
- Horizontal asymptote: The degrees in \(\frac{x+1}{x-3}\) are equal, so the horizontal asymptote is the ratio of leading coefficients, \(y=1\).
Example summary: Domain excludes \(x=1\) and \(x=3\); hole at \((1,-1)\); vertical asymptote \(x=3\); x-intercept \((-1,0)\); y-intercept \((0,-\frac{1}{3})\); horizontal asymptote \(y=1\).
5) Visualization: Graph Skeleton for the Example Rational Function
6) Checklist for Any Rational Function
- Write the function as \(f(x)=\frac{P(x)}{Q(x)}\) and factor \(P\) and \(Q\) when possible.
- Find the domain by excluding all solutions to \(Q(x)=0\).
- Cancel common factors to reveal holes; the canceled input stays excluded from the domain, and its missing y-value comes from the simplified form.
- Find vertical asymptotes from the simplified denominator \(Q_s(x)=0\).
- Find intercepts: x-intercepts from the simplified numerator \(P_s(x)=0\), and the y-intercept from \(f(0)\) if defined.
- Determine end behavior using degrees (horizontal asymptote) or long division (slant asymptote).
A rational function is a ratio of polynomials whose denominator restrictions define the domain and create holes or vertical asymptotes, while degree comparisons control end behavior through horizontal or slant asymptotes.