Sketch a graph for a function which has domain $(0,4)$ and range $(-\infty , \infty)$

Question

I tried questions like these by putting the values of range and domain in the linear function $ax+b$ type;they were closed intervals. Now I don't know how to proceed further. That approach is not working here. If anyone can suggest that would be great help thanks.

Answer 1

Take the function $\tan\left(\dfrac{\pi}{4}x-\dfrac{\pi}{2}\right)$ for $x \in (0,4)$.

I know that $\tan(x)$ has range $(-\infty, \infty)$ for $x \in \left(\frac{-\pi}{2},\frac{\pi}{2}\right)$. So if I get an arbitrary interval for the domain, say $(a,b)$ where $a \neq b$, then I just define a linear function $$f(x)=\frac{\pi}{b-a}(x-b)+\frac{\pi}{2}$$ We see that $f(a)=\frac{-\pi}{2}$ and $f(b)=\frac{\pi}{2}$ hence the function $\tan(f(x))$ has range $(-\infty, \infty)$ on $(a,b)$

Answer 2

Alternatively, take the function $\dfrac1{4-x}-\dfrac1x$ for $x\in(0,4)$.

Note that it approaches $-\infty$ as $x\to0$, it approaches $\infty$ as $x\to4$, and it is continuous.