Question

Find the domain and range of $y = \tan x$.

Answer 1

Hint: The domain is the set of all possible values of \[x\] which will satisfy the function, and thus will give us the output values of $y$. The range of a function is the complete set of all possible resulting values of the output variable $y$ which is obtained after substituting the domain.Now using the above definitions we can find the domain and range of $y = \tan x$.

Complete step by step answer:
Given, $y = \tan x.....................................\left( i \right)$.
Now we need to find the domain and range $y = \tan x$.
Now we know that by using one of the basic trigonometric identity we can write (i) as:
$\tan x = \dfrac{{\sin x}}{{\cos x}}..............................\left( {ii} \right)$
Substitute it in (i):
$y = \tan x = \dfrac{{\sin x}}{{\cos x}}.....................................\left( {iii} \right)$
Now first let’s find its domain:
So domain basically represents the values the variable \[x\] can take and thus define the function.So here we have $y = \tan x = \dfrac{{\sin x}}{{\cos x}}$ which has the denominator $\cos x$ .We also know that for a fraction the denominator must not be zero such that in this case we can say:
$\cos x \ne 0$.
Now we can find values of\[x\]:
\[\Rightarrow \cos x \ne 0 \\
\Rightarrow x \ne n\dfrac{\pi }{2}................................\left( {iv} \right) \\ \]
So on analyzing (iv) we can say that the input variable \[x\] can take any value except $n\dfrac{\pi }{2}\;{\text{where n is an odd integer}}$ .
So the domain can be written as \[\left( { - \infty ,\infty } \right) - \left( {n\dfrac{\pi }{2}} \right)\], ${\text{where n is an odd integer}}$
Now first let’s find its range: So range basically represents the values the variable \[y\] can take and thus define the function.So here we have
$y = \tan x = \dfrac{{\sin x}}{{\cos x}}$
Now we have our domain \[\left( { - \infty ,\infty } \right) - \left( {n\dfrac{\pi }{2}} \right)\],${\text{where n is an odd integer}}$
So on substituting the domain values in the given function we can obtain any value since $\tan x$ can take any value.
So the range can be written as: $\left( { - \infty ,\infty } \right)$
Therefore the domain and range of the function $y = \tan x$ is:
Domain: \[\left( { - \infty ,\infty } \right) - \left( {n\dfrac{\pi }{2}} \right)\] ${\text{where n is an odd integer}}$
Range: $\left( { - \infty ,\infty } \right)$

Note:While doing similar questions one should know the basic trigonometric conversions for ease of calculations. Also while finding the domain one should take care that the value of the denominator of the fraction to not to be zero as well as the number under a square root sign to be positive.