Question

Find the period f(x)=-tan(1/3(x-pi/4))

Answer 1

To find the period of the function f(x) = -tan(1/3(x-pi/4)), we need to find the value of a in the equation f(x+a) = f(x).

Let's plug in x = x + a into the function:

f(x+a) = -tan(1/3(x+a-pi/4))

Now, we can set this equal to f(x):

-tan(1/3(x+a-pi/4)) = -tan(1/3(x-pi/4))

Since the tangent function has a period of π, we can rewrite the above equation as:

1/3(x+a-pi/4) - x + pi/4 = nπ for some integer n

Simplifying this equation:

1/3(x+a-pi/4) - x + pi/4 = nπ

1/3(x + a - pi/4 - 3x + 3pi/4) = nπ

(1/3)(3a - 8pi/4) = nπ

3a - 8pi/4 = 3nπ

3a - 2pi = 3nπ

3a = 3nπ + 2pi

a = nπ + 2pi/3

Therefore, the period of the function f(x) = -tan(1/3(x-pi/4)) is 2pi/3.