For y = -tan x, the domain is all values that x can take so that the value of y is defined. The range is all values that y can take.
To calculate y = -tan x, we have to calculate tan x and multiply it by -1. The function is defined wherever tan x is defined. Now tan x is defined for all values of x except odd-integer multiples of pi/2, as where x is an odd integer multiple of pi/2, the value of tan x is not defined. This gives the domain as all values except odd integer multiples of pi/2.
The range is the set of all real values. Or y = [-inf, +inf].
Since the tangent function is an odd function, we'll re-write the given function as:
-tan x = tan (-x)
the domain of tangent function is (-pi/2 ; pi/2) or all real numbers, except pi/2 + kpi, k belongs to Z set of numbers.
For x = pi/2 + kpi, the tangent function is undefined.
Let's see why:
The tangent function is a ratio, where numerator is sine function and denominator is represented by cosine function:
tan x = sin x/cos x
If the denominator is zero, then the ratio is undefined. Since the cosine function is cancelling for x = pi/2 + kpi, then the ratio, namely tangent function, is undefined.
The range of tangent function is formed from all real numbers.
f(x) : (-pi/2 ; pi/2) -> R
f(x) = tan x