3.6 Rational Functions
Ideas in this lesson
- Use tables to distinguish one-sided and end behavior
- Transform reciprocal graphs and give exact domains and ranges
- Determine rational intercepts and asymptotes from factors and degrees
- Read rational graphs and reconstruct expressions
- Graph simple and repeated rational zeros and poles with exact ranges
- Distinguish holes from poles and track missing outputs
- Find slant asymptotes by division and sketch the curve
- Compare polynomial end behavior and locate numerical extrema
- Interpret saturation and limiting levels in rational models
- Analyze transient rise-and-fall models and threshold times
- Derive and interpret lens, sound, escape and resistance models
- Recognize reciprocal-square transformations and exact ranges
- Construct functions satisfying specified asymptotes and intercepts
- Prove absence of linear asymptotes and test crossing claims
Use tables to distinguish one-sided and end behavior · Worked example
For \(r(x)=\dfrac{4}{x - 3}\), complete a table at inputs \(\dfrac{29}{10}\), \(\dfrac{299}{100}\), \(\dfrac{301}{100}\), \(\dfrac{31}{10}\), \(-100\), \(100\). Use it to describe both one-sided behaviors at \(3\) and predict the horizontal asymptote. Explain why finite table values support, but do not prove, an infinite limit. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
Answer: Table outputs in the given order: \(-40\), \(-400\), \(400\), \(40\), \(- \dfrac{4}{103}\), \(\dfrac{4}{97}\). Left behavior \(-\infty\); right behavior \(\infty\). Horizontal asymptote \(y=0\).
Substitute each input without rounding: \(-40\), \(-400\), \(400\), \(40\), \(- \dfrac{4}{103}\), \(\dfrac{4}{97}\).
The factor \(x - 3\) changes sign across the excluded input.
Thus the left and right limits are \(-\infty\) and \(\infty\).
As the absolute input grows, the fractional term tends to zero; the exact end limit is \(0\).
A finite table samples only finitely many inputs and finite outputs. It cannot establish what happens at every input arbitrarily close to the excluded value, or prove that the outputs exceed every bound. Here the algebra supplies that proof: the nonzero constant numerator divided by a denominator approaching zero has unbounded magnitude, while the denominator sign determines the one-sided direction.
Written-work review:
The factor \(x - 3\) changes sign across the excluded input.
Thus the left and right limits are \(-\infty\) and \(\infty\).
As the absolute input grows, the fractional term tends to zero; the exact end limit is \(0\).
A finite table samples only finitely many inputs and finite outputs. It cannot establish what happens at every input arbitrarily close to the excluded value, or prove that the outputs exceed every bound. Here the algebra supplies that proof: the nonzero constant numerator divided by a denominator approaching zero has unbounded magnitude, while the denominator sign determines the one-sided direction.
Written-work review:
- Show the stated algebra and preserve every excluded input.
- Include all requested graph, sign-table, model or explanation components.
- Compare the written work with the supplied reference solution; selecting a statement alone does not establish mastery.