Rational expressions and the parent function 1/x
Rational Expressions and the Parent 1/x
Rational Expressions and the Parent 1/x is the opening topic of Rational Functions in the Common Core State Standards. It is aligned to Standard F-IF.7d, which requires students to graph rational functions, identifying zeros and asymptotes.
A rational function is a ratio of polynomials; the parent \(\dfrac1x\) has a vertical asymptote at \(x=0\) and a horizontal asymptote at \(y=0\).
Theory
A rational function is a ratio of polynomials \(\dfrac{p(x)}{q(x)}\). The simplest is the reciprocal parent \(f(x)=\dfrac1x\), with:
- Domain \(x\neq0\), range \(y\neq0\).
- A vertical asymptote at \(x=0\).
- A horizontal asymptote at \(y=0\).
The reciprocal parent:
How to read a reciprocal graph
- Find where the denominator is zero — the vertical asymptote.
- Find the horizontal asymptote from the transformation \(k\).
- State the domain (exclude the asymptote \(x\)).
- Sketch the two branches.
The function is undefined at \(x=0\) and never outputs \(0\).
| \(\text{domain}\) | \(=\) | \(x\neq0\) |
| \(\text{range}\) | \(=\) | \(y\neq0\) |
Dividing by a tiny positive number gives huge outputs.
| \(x\to0^+\) | \(\Rightarrow\) | \(\dfrac1x\to+\infty\) |
The denominator is zero at \(x=2\).
| \(x-2\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(2\) |
Substitute \(x=4\).
| \(f(4)\) | \(=\) | \(\dfrac14\) |
Common pitfalls
Frequently asked questions
What is a rational function?
A ratio of two polynomials, \(\dfrac{p(x)}{q(x)}\).
What are the asymptotes of \(\dfrac1x\)?
Vertical \(x=0\) and horizontal \(y=0\).
What is the domain of \(\dfrac1x\)?
All real numbers except \(0\).
Does the graph touch its asymptotes?
No — it only approaches them.