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I would like to know how you can find the range of this function $f(x)=x+\frac{1}{x}$ through for example algebra

I know it is possible to calculate the asymptotes and stationary points and then draw the graph, but how can you calculate it otherwise?

Thanks in advance

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2 Answers

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To find the range of the function put x + 1/x =y This will reduce to x^2 +1 -yx=0
Now since x is real therefore the discriminant of the above equation should be Non Negative which gives y^2 - 4 >=0 which gives y>=2 and y<=-2. Hence the range which is y is y>=2 and y<=-2

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The function is odd, so we only have to worry about $x>0.$ Since $f(x) > x,$ it is clear that $f$ takes on arbitrarily large values. What we need is the minimum value of $f$ on $(0,\infty).$ Do you know basic calculus? If so, you find the minimum value on $(0,\infty)$ is $2$ (this occurs at $x=1$). By the intermediate value theorem, $f((0,\infty)) = [2,\infty).$ Using the oddness of $f,$ we get the full range of $f$ to be $ (-\infty,-2]\cup [2,\infty).$

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