Proving Property Of An Even Function

A function \(\mathrm{f}\left( x \right)\) is said to be even if \(\mathrm{f}\left( -x \right)=\mathrm{f}\left( x \right)\) for all \(x\in {{\mathrm{D}}_{\mathrm{f}}}\).

Let \(\mathrm{f}\left( x \right)=\frac{px+q}{x+r}\) where \(p\), \(q\) and \(r\) are real constants, \(r\ne 0\) and \(x\in \mathbb{R}\mathrm{ }\!\!\backslash\!\!\mathrm{ }\left\{ \pm r \right\}\).

Prove that if \(\mathrm{f}\left( x \right)\) is an even function on the domain \(\mathbb{R}\mathrm{ }\!\!\backslash\!\!\mathrm{ }\left\{ \pm r \right\}\), then \(\mathrm{f}\left( x \right)=k\) for some real constant \(k\) which is to be determined in terms of \(p\), \(q\) or \(r\).