We have recently started studying rational functions at school. I have learnt that a rational functi

Adriana Ayers

Adriana Ayers

Answered question

2022-06-20

We have recently started studying rational functions at school. I have learnt that a rational function is the ratio of two polynomials, i.e
p ( x ) q ( x )
My question is, what if p(x) and q(x) have a common factor (linear, quadratic, etc) ??
For example, ( x 1 ) ( x 2 ) ( x 2 ) ( x 3 )
Is it still a rational function? We haven’t studied calculus in maths yet, but we have studied a little bit of calculus in our physics classes, and from what I know, I can cancel out the common factor but the function is discontinuous at x = 2, i.e it has a hole.
So is it still a rational function?

Answer & Explanation

humbast2

humbast2

Beginner2022-06-21Added 21 answers

A rational function is defined to be any function f(x) that can be expressed as P ( x ) Q ( x ) ,, where P(x) and Q(x) are polynomials and Q(x) is not the zero polynomial.
So, for example, a linear function such as f(x)=x is a rational function (as all polynomials are) because you can express f(x) as f ( x ) = x 1 , a polynomial of degree 1 divided by a polynomial of degree 0.
Rational functions, of course, can have "holes" because they are continuous everywhere except where we would divide by zero; e.g., f ( x ) = x 1 x 1 is a rational function whose graph looks like the graph of the constant function y=1 except there is a point missing at x=1.

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