Hyperbolic Sine Function

Question

Show that $\sinh(\mathbb{R})=\mathbb{R}$

I know that $\sinh(x)=\dfrac{e^x-e^{-x}}{2}$ but I can't see how inputting the set of real numbers gets the real numbers back as $e^x \gt 0$

Answer 1

Note that $-e^{-x}<0$ and $-e^{-x} \to -\infty$ as $x\to-\infty,$ while $e^x\to0$ as $x\to-\infty.$

This function

• is everywhere increasing, as demonstrated by the fact that its derivative is everywhere positive, and

• is everywhere continuous, and

• approaches $\pm\infty$ as $x\to\pm\infty$ respectively.

Thus the intermediate value theorem can be used to show that the image of this function is all of $\mathbb R.$