Range of functions like $x^3 -\sin x$

Question

How do I find range of functions like $x^3 -5x^2 +3x - 12\sin x$ or simply $x^3 - \sin x$ mathematically?

Is there an intuitive way? Is there a smart way to graph functions like $f(x)+\sin x$? Thanks in Advance.

Answer 1

• Plotting is always a good idea to quickly get an idea of what the function looks like. I would suggest using an online tool like this or this or this. Alternatively, any computer algebra system (Maple, Matlab, Mathematica, Sage, ...) or a small python script (probably using numpy and matplotlib), or maybe a graphing calculator will do nicely.

• To then answer the question of range precisely (assuming the function is continuous), you have to

  • determine the two limits $\lim_{x\to\infty}f(x)$ and $\lim_{x\to-\infty}f(x)$
  • determine any minima/maxima of the function. This is done by differentiating the function and solving for $f'(x)=0$.

For example:

• The function $x^3-\sin(x)$ goes to $-\infty$ for $x\to-\infty$ and it goes to $\infty$ for $x\to\infty$. Therefore, the domain is all real numbers. (no need to compute any minima/maxima in this case).