# Range of a trigonometry function

Let $$f(x)=\sin^3x^2-5\sin^2x^2+3$$ What is the range of this function?

I plotted this function and from plot it is obviously that for all $x\in\mathbb{R}$, $f(x)\in[-3,3]$. Then I tried to find limit of this function at $x\to\infty$ using Wolfram Mathematica and it says that $$\lim_{x\to\infty}f(x)\in[-3,4]$$ How it is possible? Does $f(x)$ have values in interval $[-3,3]$ for small values of $x$ and in interval $[-3,4]$ for bigger values of $x$ or Wolfram Mathematica is wrong? Also, how to find range of this function without plotting it?

• Hint: Find a bound for $|f(x)|$ – Extremal Dec 5 '14 at 22:46
• Are you familiar with calculus? Have you tried finding the relative extrema? – N. F. Taussig Dec 5 '14 at 22:54
• I don't know why 4 is included in the range that Wolfram gives you, but it's still technically correct. It's just a little bigger than necessary. – Brady Gilg Dec 5 '14 at 23:30

The term $\sin x^2$ takes on all values in $[-1,1]$ and your function is $u^3-5u^2+3$ (with $u=\sin x^2$), which restricted to $[-1,1]$ has a minimum of $-3$ at $u=-1$ and a maximum of $3$ at $u=0.$ So I think you're right.
Maybe the Mathematica software got "confused" looking at $x$ near $\pm \infty,$ where the curve oscillates extremely quickly over small intervals.