Question

Find the domain of the function.

6. \(g\left( x \right) = \frac{{\bf{1}}}{{1 - tanx}}\)

Short answer

The domain of the function \(g\left( x \right) = \frac{1}{{1 - \tan x}}\) is \(\left\{ {\left. {x{\rm{ }}} \right|x \ne \frac{\pi }{4} + n\pi ,x \ne \frac{\pi }{2} + n\pi } \right\}\), where \(n\) is an integer.

Step-by-step solution

The domain of the function 

The function \(g\left( x \right) = \frac{1}{{1 - \tan x}}\) is defined when the denominator of the rational function cannot be 0.

So,\(1 - \tan x \ne 0\), or\(\tan x \ne 1\).

It is known that if\(\tan x = \tan \frac{\pi }{4}\), or\(\tan x = 1\), thegeneral solution is defined as shown below:

\(x = \frac{\pi }{4} + n\pi \)

The domain of the function

If \(\tan x \ne 1\), then the general solution is defined as shown below:

\(x \ne \frac{\pi }{4} + n\pi \)

Also, it is known that the tangent function is not defined when\(x \ne \frac{\pi }{2} + n\pi \).

Thus, the domain of the function \(g\left( x \right) = \frac{1}{{1 - \tan x}}\) is \(\left\{ {\left. {x{\rm{ }}} \right|x \ne \frac{\pi }{4} + n\pi ,x \ne \frac{\pi }{2} + n\pi } \right\}\), where \(n\) is an integer.