What is Range of $x^3 -\ cos(1/x)$

Question

What is the range of $x^3 - \cos\frac{1}{x}?$

Answer 1

I think you are confused by this part,$$\cos{\frac{1}{x}}$$ This is even function oscillating between [-1, 1].

The function graph looks like: http://www.wolframalpha.com/input/?i=cos(1%2Fx)

Answer 2

I think what @shenlei was saying is correct. If you are indeed confused about the range of $\cos(\frac{1}{x}),$ then consider that $-1\leq \cos(\frac{1}{x})\leq 1,$ for all $x\ne 0.$

If you'd like to see if a number, say $b$, is in the range of the function $f(x)=x^3,$ then you can consider $x=\sqrt[3]{b},$ so $$f(\sqrt[3]{b})=(\sqrt[3]{b})^3=b,$$ and thus $x^3$ is onto, its range is all of the real numbers $\Bbb{R},$ and is called onto.

Finally, you should ask yourself, what happens to $$x^3-\cos\left(\frac{1}{x}\right),$$ then? Well, you should try to rationalize that $x^3$ having a range of all real numbers indicates that $x^3-\cos\left(\frac{1}{x}\right)$ has the same range.