Find the Asymptotes f(x)=tan(3x-pi/2)

Problem

ƒ(x)=tan(3*x−π/2)

Solution

1. Identify the standard vertical asymptotes for the tangent function tan(θ) The function is undefined where the input θ is an odd multiple of π/2

θ=π/2+n*π

2. Set the argument of the given function, 3*x−π/2 equal to the general expression for the asymptotes of the tangent function.

3*x−π/2=π/2+n*π

3. Isolate the term containing x by adding π/2 to both sides of the equation.

3*x=π/2+π/2+n*π

4. Simplify the constant terms on the right side of the equation.

3*x=π+n*π

5. Solve for x by dividing the entire equation by 3

x=π/3+(n*π)/3

6. Factor out the common term to express the vertical asymptotes in a simplified form, where n is any integer.

x=((n+1)*π)/3

7. Determine horizontal or oblique asymptotes. Since the tangent function ranges from −∞ to ∞ and is periodic, there are no horizontal or oblique asymptotes.

Final Answer

x=π/3+(n*π)/3

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