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# Do transformed functions of the form $f(x)=a \tan \left(\frac{1}{b} x\right)$ have the same behavior seen in the parent function, that is, f(-x)=-f(x) ? Justify your answer and how it helps to graph transformed functions.

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The tangent function can be described in form

\begin{aligned} f(x)=a\tan\left(\frac{1}{b}x\right) \end{aligned}

If $\tan x=\frac{\sin x}{\cos x}$, then

\begin{aligned} \tan(-x)&=\frac{\sin(-x)}{\cos(-x)}\\ &=\frac{-\sin x}{\cos x}\\ &=-\tan x \end{aligned}

Thus tangent function takes the form $f(-x)=-f(x)$, so the tangent is an odd function.

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