Holes in Rational Graphs
A rational function is any function that can be written as a ratio of two polynomials, f(x) = P(x)/Q(x). When both the numerator and denominator share a common factor, something interesting happens at the value of x that makes that shared factor equal to zero: the function appears to be undefined there, but the discontinuity is 'removable' — meaning the graph looks like a continuous curve everywhere except for a single missing point, called a hole. Holes are formally known as removable discontinuities. Unlike vertical asymptotes — where the function shoots off to positive or negative infinity — a hole is a single isolated point missing from the graph. If you were to cancel the common factor from numerator and denominator, you would get a simpler, fully defined function. The hole exists exactly at the x-value where that cancelled factor would equal zero, and its y-coordinate is found by plugging that x-value into the simplified function. Understanding holes is essential for accurately sketching rational functions, analyzing their domains, and working with limits in pre-calculus and calculus. The concept connects algebraic factoring to graphical behavior: every common factor you cancel corresponds to precisely one hole in the graph. Recognizing the difference between a hole and a vertical asymptote prevents errors in graphing and in describing a function's domain and range.
- Factor the numerator and denominator completely
- Identify common factors
- State the domain restriction
- Cancel the common factor to get the simplified function
- Find the coordinates of each hole
- Graph the simplified function and mark the holes
Factor the numerator and denominator completely
Write P(x) and Q(x) in fully factored form. For example, if f(x) = (x^2 - x - 6)/(x^2 - 9), factor to get f(x) = (x-3)(x+2)/[(x-3)(x+3)]. Always factor before drawing any conclusions about the graph.
Identify common factors
Look for factors that appear in BOTH the numerator and the denominator. In the example, (x-3) appears in both. Each such common factor, say (x - a), signals a hole at x = a. If a factor appears only in the denominator (not cancelled), it produces a vertical asymptote instead.
State the domain restriction
Before cancelling, note every x-value that makes the original denominator zero. For f(x) = (x-3)(x+2)/[(x-3)(x+3)], the denominator is zero at x = 3 and x = -3. Both values must be excluded from the domain: Domain = {x ∈ ℝ : x ≠ 3 and x ≠ -3}.
Cancel the common factor to get the simplified function
After noting the restriction, cancel the shared factor: f(x) = (x+2)/(x+3), valid for x ≠ 3 and x ≠ -3. This simplified form is used for all calculations except at the excluded points. At x = -3, the simplified function still has a zero denominator, so x = -3 is a vertical asymptote — not a hole.
Find the coordinates of each hole
For each cancelled factor (x - a), substitute x = a into the simplified function to get the y-coordinate of the hole. In the example: y = (3+2)/(3+3) = 5/6. The hole is at the point (3, 5/6). On a graph, this is represented as an open circle at that coordinate.
Graph the simplified function and mark the holes
Draw the graph of the simplified rational function — including any remaining asymptotes and intercepts — but place an open circle (hollow dot) at every hole. The curve passes through every other point normally; only the holes are 'punched out' of the graph.
The Key Equation
- P(x) — Numerator polynomial (dimensionless): The polynomial in the numerator; must be factored to find common factors.
- Q(x) — Denominator polynomial (dimensionless): The polynomial in the denominator; zeros of Q(x) are candidates for holes or vertical asymptotes.
- (x-a) — Common factor (dimensionless): The factor shared by both P(x) and Q(x); its zero x = a is the x-coordinate of the hole.
- R(x)/S(x) — Simplified function (dimensionless): The reduced form after cancellation; used to compute the y-coordinate of the hole by substituting x = a.
Worked Example
Problem. Find all holes and vertical asymptotes of f(x) = (x^2 + x - 12) / (x^2 - 2x - 3), then state the coordinates of each hole.
Answer. Hole at (3, 7/4); vertical asymptote at x = -1; horizontal asymptote at y = 1.
Origins · 1 minAugustin-Louis Cauchy and how this was discovered
Cauchy was the first mathematician to give a rigorous definition of a limit, which is the formal tool used to explain why holes (removable discontinuities) behave differently from other types of discontinuities. He defined continuity precisely and showed how the limit of a function at a point can exist even if the function is undefined there — exactly the situation at a hole in a rational graph. Cauchy published an astonishing 789 mathematical papers and books during his lifetime, averaging nearly one new publication every single week for decades.
Related Concepts
- Vertical Asymptotes of Rational Functions — Vertical asymptotes and holes both arise from zeros of the denominator; distinguishing between them requires identifying whether the zero is also a zero of the numerator.
- Domain of Rational Functions — Holes remove individual points from the domain; understanding domain restrictions is the first step in locating holes.
- Factoring Polynomials — Finding holes requires completely factoring both numerator and denominator to identify common factors.
- Introduction to Limits — A hole is exactly the situation where the limit of the function exists at a point but the function value does not, making holes a gateway to understanding limits.
- Graphing Rational Functions — Holes are one of several key features — along with asymptotes and intercepts — needed to produce an accurate graph of a rational function.