The tangent function is periodic. It has a period of pi. The function from – pi/2 to pi/2 repeats itself again and again.
The tangent function is an odd function. That means it is symmetric to the origin. In algebraic terms this means that tan(-x) = – tan(x).
It forms an S shaped curve that goes through the origin. The function has discontinuities at multiples of pi/2. It has vertical asymptotes there. From the left it approaches positive infinity. From the right it approaches negative infinity.
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