# What is the domain and range for y = 6sin^-1(4x)?

Jul 28, 2015

domain : $- \frac{1}{4} \le x \le \frac{1}{4}$

range : $y \in \mathbb{R}$

#### Explanation:

Remember simply that the domain of any function are the values of $x$ and the range is the set of values of $y$

Function : $y = 6 {\sin}^{-} 1 \left(4 x\right)$

Now, rearrange our function as : $\frac{y}{6} = {\sin}^{-} 1 \left(4 x\right)$

The corresponding $\sin$ function is $\sin \left(\frac{y}{6}\right) = 4 x$ then $x = \frac{1}{4} \sin \left(\frac{y}{6}\right)$

Any $\sin$ function oscillates between $- 1$ and $1$

$\implies - 1 \le \sin \left(\frac{y}{6}\right) \le 1$

$\implies - \frac{1}{4} \le \frac{1}{4} \sin \left(\frac{y}{6}\right) \le \frac{1}{4}$

$\implies - \frac{1}{4} \le x \le \frac{1}{4}$

Congratulations you have just found the domain(the values of $x$)!

Now we proceed to find the values of $y$.

Starting from $x = \frac{1}{4} \sin \left(\frac{y}{6}\right)$

We see that any real value of $y$ can satisfy the above function.

Meaning that $y \in \mathbb{R}$