Which of the following represent the domain and range of y = tan x

The correct option is the third one.

Step-by-step explanation:

We have the function y = tan(x)

and we want to find the domain and range, where the domain is the set of numbers that can be x, and the domain is the possible values of y.

Tan(x) = sin(x)/cos(x)

for the domain, we can see in which values of x we have problems. The problems are when the denominator is equal to zero, so when x = pi/2, 3*pi/2, etc, we have problems.

So x can not be equal to ((2n + 1)/2)*pi = pi/2 + n*pi

Now, for the range, we have that when cos(x) is near zero, sin(x) is near 1, so near this point, the tan(x) will tend to infinity or -infinity. and tan(x) is a differentiable, so it is continuous in its domain, then all the values in between are also in the range, then the range is all the real numbers.

now we have:

Domain: X is real and different than pi/2 + n*pi for n an integer number.

Range: Y can be any real number.

The correct option is the third one.

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!

42

step-by-step explanation: