Determine which functions are rational

[chwala]

Homework Statement

consider the attached problem

Relevant Equations

rational functions

Ok in my thinking, i would say that it depends on $x$, if $x$ belongs to the integer class, then the rational functions would be $i$ and $iii$...but from my reading of rational functions, i came up with this finding:

I would appreciate your input on this.

 

[Delta2]

Well according to the above definition only (i) and (iii) are rational functions.

(ii) or (iv) doesn't look like they are a fraction of polynomials neither I can find a way to transform them to a fraction of two polynomials. For example if we multiply both the numerator and denominator of (iv) by $\sqrt{x-1}$ we get $$\frac{x-1}{\sqrt{x^2-1}}$$ the numerator has become polynomial, but the denominator still isn't polynomial.

 

[chwala]

Well according to the above definition only (i) and (iii) are rational functions.

(ii) or (iv) doesn't look like they are a fraction of polynomials neither I can find a way to transform them to a fraction of two polynomials. For example if we multiply both the numerator and denominator of (iv) by $\sqrt{x-1}$ we get $$\frac{x-1}{\sqrt{x^2-1}}$$ the numerator has become polynomial, but the denominator still isn't polynomial.

thanks, i guess i missed the term "polynomial" cheers mate