Y= floor(x)-x+2

from the graph, it is periodic. but how to prove it?

It is sufficient to prove that is periodic. We claim that the period is 1, and here's the proof:

Let

by definition of periodic function, we are claiming that, for every value of x,

The floor is additive with integers: if a is a real number and b is an integer, we have

So, we have

Which proves that the function is periodic with period 1. The idea is that is the decimal part of x. For example, if you start at x=2.5 you have

If you go one period ahead, i.e. if you add 1, you have

And as you can see, the integer part doesn't matter, because you'll always have a difference in the form

xx - xx.5 = -0.5

it happens during combustion and impacts the environment because the plants and trees take in that and it can cause them to die.

p ÷ r = tan(x)

step-by-step explanation:

given that there is a figure below shows a right triangle: a right triangle is shown with hypotenuse equal to q units and the length of the legs equal to p units and r units.

graph is missing so i will create the graph as per given information.

the angle between the legs having lengths p units and q units is y degrees. the angle between the legs having lengths r units and q units is x degrees.

now we need to find about what is p ÷ r equal to.

so we can use trigonometric ratio tan(x)=[altitude]/[base] to find the value of p ÷ r, because p is the altitude and r is the base.

p ÷ r=tan(x)

hence final answer is p ÷ r=tan(x).