Rational Function

1. Concept Overview

A rational function is a function that can be written as a ratio of two polynomials.

These functions are extremely important because they:

• introduce restrictions in domain,
• show breaks and asymptotes,
• and are widely used in limits and graph analysis.

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2. Definition of Rational Function

A function [f] is called a rational function if

[f(x)=\dfrac{p(x)}{q(x)}]

where:

• [p(x)] and [q(x)] are polynomial functions
• [q(x)≠0]

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3. Examples of Rational Functions

• [f(x)=\dfrac{1}{x}]
• [g(x)=\dfrac{x+1}{x−2}]
• [h(x)=\dfrac{x²−1}{x+3}]

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4. Domain of a Rational Function

Important Rule:
A rational function is not defined when its denominator becomes zero.

Steps to find domain:

1. Equate denominator to zero
2. Exclude those values from real numbers

Example:
[f(x)=\dfrac{1}{x−3}]

Denominator [x−3=0 ⇒ x=3]

Domain:
[(−∞,3)∪(3,∞)]

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5. Simplification and Domain

Even if factors cancel during simplification, the restricted values remain excluded.

Example:
[f(x)=\dfrac{x−1}{x−1}]

Though it simplifies to [1],
[x=1] is not allowed.

Domain:
[(−∞,1)∪(1,∞)]

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6. Nature of Rational Functions

A rational function may be:

• one–one or many–one
• continuous in parts
• discontinuous at points where denominator is zero

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7. Graphical Behaviour

• May have vertical asymptotes
• May have horizontal or oblique asymptotes
• Graph may have breaks

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8. Conceptual Questions with Solutions

1. What is a rational function?

A rational function is a function expressed as a ratio of two polynomials.

2. Why must denominator not be zero?

Because division by zero is undefined.

3. Is every polynomial function a rational function?

Yes. Any polynomial can be written as a rational function with denominator 1.

4. Is every rational function a polynomial?

No. Presence of variable in denominator makes it non-polynomial.

5. Can rational functions be discontinuous?

Yes. They are discontinuous where the denominator is zero.

6. Does simplification change the domain?

No. Restricted values remain excluded.

7. Can a rational function be continuous everywhere?

Yes, if the denominator is never zero.

8. Why do rational functions have asymptotes?

Because the function may approach a fixed value but never reach it.

9. Can rational function be one–one?

Yes. Some rational functions are one–one.

10. Why are rational functions important in calculus?

They help in understanding limits, discontinuity, and asymptotic behaviour.

11. Is [f(x)=1/x] a rational function?

Yes. Both numerator and denominator are polynomials.

12. Can denominator be a constant?

Yes. It still remains a rational function.

13. Why is domain of rational function restricted?

Because some values make the denominator zero.

14. Is [f(x)=\dfrac{x}{x}] defined for all x?

No. It is undefined at x=0.

15. Can rational functions have holes in graph?

Yes. Removable discontinuities create holes.

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9. FAQ / Common Misconceptions

1. Cancelled terms remove restrictions.

False. Restrictions remain.

2. Rational functions are always discontinuous.

False. Some are continuous everywhere.

3. Domain is always R.

False. Domain excludes denominator zeros.

4. Rational functions have no graph.

False. They have well-defined graphs.

5. Asymptotes are part of graph.

False. They are never touched.

6. Rational functions cannot be one–one.

False. Some are one–one.

7. Rational functions are not useful.

False. They are very important.

8. Simplified form gives domain.

False. Always use original expression.

9. All rational functions have asymptotes.

False. Some do not.

10. Rational functions are polynomial.

False. They are different.

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10. Practice Questions

Question 1.

Find the domain of [f(x)=\dfrac{1}{x−4}].

Step-by-Step Solution:

1. Denominator becomes zero at [x=4].
2. Exclude [4].

Conclusion:
Domain = [(−∞,4)∪(4,∞)]

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Question 2.

Find the domain of [f(x)=\dfrac{x+2}{x²−9}].

Step-by-Step Solution:

1. Denominator [x²−9=0 ⇒ x=±3].
2. Exclude [−3] and [3].

Conclusion:
Domain = [(−∞,−3)∪(−3,3)∪(3,∞)]

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Question 3.

State whether [f(x)=\dfrac{1}{x}] is one–one.

Step-by-Step Solution:

1. Different inputs give different outputs.

Conclusion:
The function is one–one.

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Question 4.

Find the domain of [f(x)=\dfrac{x}{x−1}].

Step-by-Step Solution:

1. Denominator zero at [x=1].

Conclusion:
Domain = [(−∞,1)∪(1,∞)]

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Question 5.

Is [f(x)=\dfrac{x−2}{x−2}] defined for all real x?

Step-by-Step Solution:

1. Denominator zero at [x=2].
2. Simplification does not remove restriction.

Conclusion:
Not defined at [x=2].