Question 3, Exercise 9.1

Solutions of Question 3 of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

Find domain and range: $y=7 \cos 4x$

Solution.

AS \begin{align*} & -1\leq \cos 4x \leq 1 \,\, \forall \,\, x\in \mathbb{R} \\ \implies & -7\leq 7 \cos 4x \leq 7 \\ \end{align*} Thus domain $= ]-\infty, \infty[ = \mathbb{R}$

Range $=[-7,7]$.

Find domain and range: $y=\cos \frac{x}{3}$

Solution.

AS \begin{align*} & -1\leq \cos \frac{x}{3} \leq 1 \,\, \forall \,\, x\in \mathbb{R} \\ \end{align*} Thus domain $= ]-\infty, \infty[ = \mathbb{R}$

Range $=[-1,1]$.

Find domain and range: $y=\sin \frac{2 x}{3}$

Solution.

AS \begin{align*} & -1 \leq \sin \frac{2x}{3} \leq 1 \,\, \forall \,\, x \in \mathbb{R} \end{align*} Thus domain $= ]-\infty, \infty[ = \mathbb{R}$

Range $=[-1, 1]$.

Find domain and range: $y=7 \cot \frac{\pi}{2} x$

Solution.

Let $\theta=\frac{\pi}{2} x$. Then $$y=7 \cot \theta$$

Domain of $y=\left\{\theta: \theta\in \mathbb{R} \text{ and } \theta \neq n\pi, n\text{ is integer} \right\}$

Range of $y=\mathbb{R}$

As \begin{align*} & \theta \neq n\pi \\ \implies & \dfrac{\pi}{2} x \neq n\pi \\ \implies & x \neq 2n \end{align*}

Hence domain of $y=\left\{x: x\in \mathbb{R} \text{ and } x \neq 2n, n\text{ is integer} \right\}$

Range of $y=\mathbb{R}$.

Find domain and range: $y=4 \tan \pi x$.

Solution.

Let $\theta=\pi x$. Then $$y=4 \tan \theta$$

Domain of $y=\left\{\theta: \theta\in \mathbb{R} \text{ and } \theta \neq (2n+1)\frac{\pi}{2}, n\text{ is integer} \right\}$

Range of $y=\mathbb{R}$

As \begin{align*} & \theta \neq (2n+1)\frac{\pi}{2} \\ \implies & \pi x \neq (2n+1)\frac{\pi}{2} \\ \implies & x \neq \frac{2n+1}{2} \end{align*}

Hence domain of $y=\left\{x: x\in \mathbb{R} \text{ and } x \neq \frac{2n+1}{2}, n\text{ is integer}\right\}$

Range of $y=\mathbb{R}$.

Find domain and range: $y=\operatorname{Cosec} 4 x$

Solution.

Let $\theta=4 x$. Then $$y= \operatorname{Cosec} \theta$$

Domain of $y=\left\{\theta: \theta\in \mathbb{R} \text{ and } \theta \neq n\pi, n\text{ is integer} \right\}$

Range: $y\leq -1 \text{ and } y\geq 1$.

As \begin{align*} & \theta \neq n\pi \\ \implies & 4 x \neq n\pi \\ \implies & x \neq \frac{n \pi}{4} \end{align*}

Hence domain of $y=\left\{x: x\in \mathbb{R} \text{ and } x \neq \frac{n \pi}{4}, n\text{ is integer} \right\}$

Range: $y\leq -1 \text{ and } y\geq 1$.

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