Question

What is the domain and range of a rational function?

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Solution

$\mathrm{Rational}\mathrm{function},f\left(x\right)=\frac{P\left(x\right)}{Q\left(x\right)}\mathrm{where}P\left(x\right)\mathrm{and}Q\left(x\right)\mathrm{are}\mathrm{polynomial}\mathrm{and}Q\left(x\right)\ne 0.\phantom{\rule{0ex}{0ex}}\mathrm{Therefore},\mathrm{domain}\mathrm{of}f\left(x\right)i\mathrm{s}\mathrm{all}\mathrm{real}\mathrm{numbers}\mathrm{except}\mathrm{those}\mathrm{values}\mathrm{of}x\mathrm{for}\mathrm{which}Q\left(x\right)=0.\phantom{\rule{0ex}{0ex}}\mathrm{The}\mathrm{range}\mathrm{of}\mathrm{rational}\mathrm{function}\mathrm{is}\mathrm{different}\mathrm{for}\mathrm{different}\mathrm{functions}.\phantom{\rule{0ex}{0ex}}\mathrm{Ex}:f\left(x\right)=\frac{1}{{x}^{2}}\phantom{\rule{0ex}{0ex}}\mathrm{Domain}:\phantom{\rule{0ex}{0ex}}{x}^{2}\mathrm{is}0\mathrm{for}x=0\phantom{\rule{0ex}{0ex}}\mathrm{Therefore},\mathrm{domain}\mathrm{of}\mathrm{f}\left(\mathrm{x}\right)\mathrm{is}R-\left\{0\right\}\phantom{\rule{0ex}{0ex}}\mathrm{Range}:\phantom{\rule{0ex}{0ex}}\mathrm{The}\mathrm{value}\mathrm{of}{x}^{2}\mathrm{will}\mathrm{always}\mathrm{be}\mathrm{positive}\mathrm{and}\mathrm{so}\frac{1}{{x}^{2}}\mathrm{will}\mathrm{also}\mathrm{be}\mathrm{always}\mathrm{positive}.\phantom{\rule{0ex}{0ex}}\mathrm{Hence},\mathrm{range}\mathrm{of}f\left(x\right)\mathrm{is}\left(0,\infty \right)\phantom{\rule{0ex}{0ex}}$

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