# For what values of x, if any, does #f(x) = tan((pi)/4-9x) # have vertical asymptotes?

Period:

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asymptotes

graph{y-tan(0.7854-9x)=0 [-5, 5, -2.5, 2.5]}

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The function f(x) = tan((pi)/4-9x) has vertical asymptotes when the tangent function is undefined. The tangent function is undefined when the angle is equal to (2n+1)(pi)/2, where n is an integer.

Setting (pi)/4-9x equal to (2n+1)(pi)/2 and solving for x, we get:

(pi)/4-9x = (2n+1)(pi)/2 -9x = (2n+1)(pi)/2 - (pi)/4 x = [(2n+1)(pi)/2 - (pi)/4]/(-9)

Therefore, the function f(x) = tan((pi)/4-9x) has vertical asymptotes at x = [(2n+1)(pi)/2 - (pi)/4]/(-9), where n is an integer.

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