Answer: 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.
