It depends on the domain of x.
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The period of the tangent function is PI. The period of y= tan(2x) is PI over the coefficient of x = PI/2
tan(2x) = 2tanx/(1 - tan2x) So tan(2x) +1 = 2tanx/(1 - tan2x) + 1 = 2tanx/(1 - tan2x) + (1 - tan2x)/(1 - tan2x) = (-tan2x + 2tanx + 1)/(1 - tan2x)
The range of y=2x is All Real Numbers
y=|x|/4 The range is [0 , ∞ )
The range depends on the domain, which is not specified.