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 Texas Essential Knowledge and Skills (Algebra II, §111.40). It is aligned to Standard 2A.6(G), which requires students to analyze the effect on the graph of changing parameters of a rational function, including its asymptotic behavior.
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.