Question

Assertion (A): Every rational function is continuous at every real number \(x\) at which it is defined.
Reason (R): Every rational function is a quotient of two polynomials.

• A. Both A and R are true, and R is the correct explanation of A
• B. Both A and R are true, but R is not the correct explanation of A
• C. A is true, but R is false
• D. A is false, but R is true

Hint:

For continuity of rational functions, just check where the denominator is non-zero.

Answer 1

The Correct Option is A

A rational function is defined as the ratio of two polynomials: \[ f(x) = \frac{p(x)}{q(x)} \] Polynomials are continuous everywhere, and the quotient of two continuous functions is also continuous at every point where the denominator is not zero. Therefore, a rational function is continuous at all \(x\) where it is defined (i.e., where \(q(x) \ne 0\)). So, the assertion is true. The reason correctly explains this: the structure of a rational function (being a quotient of polynomials) guarantees continuity at all defined points.