Question

Find the domain of the function y = 3 tan(23x)

Answer 1

`\mathbb{R} \backslash \displaystyle \left\lbrace \left. (1)/(23)\, \left(k\, \pi + (\pi)/(2)\right) \; \right| k \in \mathbb{Z} \right\rbrace`.

In other words, the
`x` in
`f(x) = 3\, \tan(23\, x)` could be any real number as long as
`x \\e \displaystyle (1)/(23)\, \left(k\, \pi + (\pi)/(2)\right)` for all integer
`k` (including negative integers.)

Explanation:

The tangent function
`y = \tan(x)` has a real value for real inputs
`x` as long as the input
`x \\e \displaystyle k\, \pi + (\pi)/(2)` for all integer
`k`.

Hence, the domain of the original tangent function is
`\mathbb{R} \backslash \displaystyle \left\lbrace \left. \left(k\, \pi + (\pi)/(2)\right) \; \right| k \in \mathbb{Z} \right\rbrace`.

On the other hand, in the function
`f(x) = 3\, \tan(23\, x)`, the input to the tangent function is replaced with
`(23\, x)`.

The transformed tangent function
`\tan(23\, x)` would have a real value as long as its input
`(23\, x)` ensures that
`23\, x\\e \displaystyle k\, \pi + (\pi)/(2)` for all integer
`k`.

In other words,
`\tan(23\, x)` would have a real value as long as
`x\\e \displaystyle (1)/(23) \, \left(k\, \pi + (\pi)/(2)\right)`.

Accordingly, the domain of
`f(x) = 3\, \tan(23\, x)` would be
`\mathbb{R} \backslash \displaystyle \left\lbrace \left. (1)/(23)\, \left(k\, \pi + (\pi)/(2)\right) \; \right| k \in \mathbb{Z} \right\rbrace`.