Key takeaway: A rational function is a function written as one polynomial divided by another, f(x) = P(x)/Q(x), whose graph often has breaks and asymptotes where the denominator is zero.
Grade 11 – Algebra 2F-IF.C.7dFunctions and Graphs
Definition of Rational Function
A rational function is a rule that divides one polynomial by another. Its inputs are every real number except the places where the bottom polynomial vanishes, so the graph usually splits into separate branches.
Close to a forbidden input the outputs grow huge, and the curve hugs a vertical line it never touches. Far out to the left and right, the curve levels off toward a horizontal line or follows a slanted one, depending on the degrees of the top and bottom. Reading those guide lines is the main skill with any rational function.
Formula and notation
The standard form is
\[f(x)=\frac{P(x)}{Q(x)}, \quad Q(x)\neq 0,\]
with \(P\) and \(Q\) polynomials. The domain leaves out every root of \(Q\). Zeros of \(f\) come from \(P(x)=0\), and the y-intercept is \(f(0)\).
Key facts
- The domain is all real numbers except the roots of the denominator.
- Equal degrees on top and bottom give a horizontal asymptote at the ratio of leading coefficients.
- Zeros of the numerator that are not zeros of the denominator are x-intercepts.
- A factor shared by top and bottom leaves a hole, not an asymptote.
Rational Function example
A graphing class studies \(f(x)=\frac{2x+3}{x-4}\). Find its domain, both asymptotes and both intercepts.
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The denominator vanishes only at 4. So 4 is missing from the domain, and the dashed vertical line sits at \(x=4\).
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Top and bottom each have degree 1, so the level line is the quotient of the leading coefficients, \(\frac{2}{1}\), which is \(y=2\).
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The numerator \(2x+3\) is zero at \(x=-\frac{3}{2}\), giving the x-intercept.
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Input 0 gives \(\frac{3}{-4}\), so the y-intercept is \(-\frac{3}{4}\).
Every real number except 4 is allowed; the asymptotes are \(x=4\) and \(y=2\); the intercepts are \((-\frac{3}{2},0)\) and \((0,-\frac{3}{4})\).
Common mistake
Watch out
Jonas reads the denominator \(x-4\) and draws the vertical dashed line at \(x=-4\).
Why it is wrong
The line goes where the denominator equals zero, so solve \(x-4=0\) instead of copying the constant with its sign. At \(x=-4\) the denominator is \(-8\), so this rational function is perfectly defined there.
Related terms and lessons
Browse the glossary: A–Z list · Algebra 2 vocabulary · Functions and Graphs