Question

Find the domain and the range of each of the following functions :

(i) $f(x)=\frac{1}{\sqrt{4+3 \sin x}}$

(ii) $f(x)=x$ !

(iii) $f(x)=\frac{x^2-9}{x-3}$

(iv) $f(x)=\sin ^2\left(x^3\right)+\cos ^2\left(x^3\right)$

Answer 1

(i) $f(x)=\frac{1}{\sqrt{4+3 \sin x}}$

Domain $4+3 \sin x>0 \Rightarrow \sin x>-\frac{4}{3}$ Always true $\Rightarrow x \in R$

Range $-3 \leq 3 \sin x \leq 3$

$\Rightarrow 1 \leq 4+3 \sin x \geq 7 \Rightarrow 1 \geq \frac{1}{4+3 \sin x} \geq \frac{1}{7} \Rightarrow 1 \geq \frac{1}{\sqrt{4+3 \sin x}} \geq \frac{1}{\sqrt{7}} \Rightarrow y \in\left[\frac{1}{\sqrt{7}}, 1\right]$

(ii) $f(x)=x$ I

Domain $x \in W \Rightarrow x \in N \cup\{0\}$

Range $\{y: y \in n !$, where $n=0,1,2,3, \ldots .$.

(iii) $f(x)=\frac{x^2-9}{x-3}=\frac{(x-3)(x+3)}{(x-3)}=(x+3)$

Domain $x \in R-\{3\}$

Range $y \in R-\{6\}$

(iv) $f(x)=\sin ^2\left(x^3\right)+\cos ^2\left(x^3\right) \\$

$f(x)=1 \\$

$\text { Domain } x \in R \\$

$\text { Range } y \in\{1\}$