Non-logarithmic bijective function from $\mathbb{R}^+$ into $\mathbb{R}$

Question

I knew 2 functions that bijectively transform $(0,\infty)$ into $(-\infty,\infty)$, which are $\log x$ and $\log\log(x+1)$. Do you guys know another function like (as simple as possible) but does not involve $\log$? Thank you so much

Answer 1

Take any positive continuous $f$ on $(0,\infty)$ such that

$$\int_0^1 f(t)\, dt = \infty=\int_1^\infty f(t)\, dt.$$

Then $F(x) = \int_1^x f(t)\, dt$ has the desired property.

Proof: By the FTC, $F'(x)=f(x)>0$ for all $x.$ Hence $F$ is strictly increasing, and therefore injective. Verify that $F(x)\to -\infty$ as $x\to 0^+,$ and $F(x)\to \infty$ as $x\to \infty.$ By the IVT, $F$ maps $(0,\infty)$ onto $\mathbb R.$

This gives lots of examples. For example, we could take $f(t) = t^{-3/2} + e^t.$ The corresponding $F(x)$ is $-2x^{-1/2} +e^x +2-e.$

Answer 2

For example, take $$f(x)=x-\frac{1}{x}.$$