Question

Find the domain and range of the function f defined by f(x) = 1/1-sinx

Answer 1

(a) The domain of a function is the set of input or argument values for which the function is real and defined.

Domain of $\frac{1}{1-sinx}$:

$$=\begin{bmatrix} solution:& 2\pi \le x \lt \frac{\pi}{2 }+2\pi n\space or\space \frac{\pi}{2 }+2\pi n\lt x\lt 2\pi \space+\space2\pi n \\ interval \space Notation & \lbrack2\pi n,\space\frac{\pi}{2} + 2\pi n\rparen\bigcup \ \lparen\frac{\pi}{2}+2\pi n,\space2\pi+2\pi n\rparen \end{bmatrix}$$

$$=2\pi n \leq x \lt \frac{\pi}{2} +2\pi n \space or \space \frac{\pi}{2} +2\pi n \lt x \lt 2\pi +2\pi n$$

The range is the set of values of the dependent variable for which a function is defined

$$Range \space of\space\frac{1}{1-sinx}:\space= \begin{bmatrix} solution: & \frac{1}{2} \leq f(x)\leq 1 \\ interval\space notation: &\lbrack \frac{1}{2},1\rbrack \end{bmatrix}$$

$$Range = \lbrack \frac{1}{2}, 1 \rbrack$$