What will be the graph for this function?

Question

There are given two functions, $$f(x)=\max(\sin t, x<t<x+1)$$

And

$$g(x)=\min(\sin t, x<t<x+1)$$

I am unable to find graphs of these functions. I've tried for a small range $x=0$ to $\pi\over{2}$. But in this small range consumed very much time taking cases. How will the graph look like?

Answer 1

I will plot the graph for $f(x)$, and the graph for $g(x)$ will be of a similar type (shifted and reflected along $x$-axis). $f(x)$ is periodic with period $2\pi$. So for $x\in[0,2\pi]$

At extremities the values will be tending to $f-$images of them.

$f(x)=\begin{cases} \sin(x+1) & 0< x<\frac{\pi}{2}-1 \\ 1 & \frac{\pi}{2}-1< x< \frac{\pi}{2} \\ \sin{x} & \frac{\pi}{2}< x<\pi+k \\ \sin(x+1) & \pi-k<x<2\pi \end{cases}$

for $0<x<\frac{\pi}{2}-1$ obviously $\sin(x+1)$ will donate. As sine function between $0, \frac{\pi}{2}$ is increasing.

Then it will remain $1$ till $\frac{\pi}{2}$

between $\frac{\pi}{2}, \pi+k$ sine function is decreasing so $\sin x$ will dominate. ($k$ will be revealed ahead)

If you look carefully at the sinusoidal graph between $\pi$ and $\frac{3\pi}{2}$ graph is decreasing and ahead of it is increasing. So there comes a point $\pi+k$ where the increasing and decreasing graph will meet, and then it follows the increasing one till $2\pi$. (you can find $k$ by $\sin(\pi+k)=\sin(\pi+1+k)$

A rough graph to give an idea is given below,