# Domains and Ranges of Standard Functions Set 3

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11. Trigonometric functions

There are six basic trigonometric functions as you know. Here, we will discuss how we can arrive at their graphs.

To draw them, we use the device of a unit circle and a point P moving in an anticlockwise direction on the circumference of this circle as shown in Fig - 18. ‘x’ is the angle in radians that OP makes with the x-axis.

We see that

\begin{align}\text{sin}x=\frac{a}{1}\end{align}=a

\begin{align}\text{cos}x=\frac{b}{1}\end{align}=b

\begin{align}\text{tan}x=\frac{a}{b}\end{align}

As the point P moves (x increases), note the variations in ‘a’ and ‘b’ and hence observe the details of the graphs we obtain.

(i) sin x (variation in ‘a’)

Domain = $$\mathbb{R}$$

Range= [–1, 1]

The graph repeats after every complete revolution of P, i.e., after every increment(or decrement) of $$2\pi$$  in the angle ‘x’. This phenomenon is known as periodicity of the graph. We say that the graph is periodic with period ‘$$2\pi$$’.

(ii) cos x (variation in ‘b’)

Domain = $$\mathbb{R}$$

Range = [–1, 1]

This function is also periodic with period $$2\pi$$.

(iii) tan x (variation in ‘a/b’)

Domain = $$\mathbb{R}$$ – \begin{align}\left\{ {n\pi + \frac{\pi }{2}} \right\}\end{align}

[Exclude the set of all points where b becomes 0, i.e. where the

angle x = n $$\pi$$ + $$\pi /2$$]

Range = $$\mathbb{R}$$

This function is periodic with period $$\pi$$. (half a revolution of P).

(iv) cosec x (variation in ‘1/a’)

$${\text{Domain}} = \mathbb{R} - \left\{ {n\pi } \right\},n \in \mathbb{Z}$$

$${\rm{Range}} = ( - \infty , - 1]\; \cup \;\;[1,\infty )$$

$$f\left( x \right) = {\rm{cosec}}\;x\;{\rm{is\;periodic\;with\;period\;2}}\pi$$

(v) sec x (variation in ‘1/b’)

\begin{align}{\text{Domain}} = \mathbb{R} - \left\{ {\left( {2n + 1} \right)\frac{\pi }{2}} \right\},\;n \in \mathbb{Z}\end{align}

[We exclude all points where cos x = 0, because \begin{align}\sec x = \frac{1}{{\cos x}}\end{align}]

$${\rm{Range}} = \left( { - \infty , - 1} \right]\; \cup \;\left[ {1,\infty } \right)\,$$

$$f\left( x \right) = \sec x\;{\rm{is\;again\;periodic\;with\;period\;2}}\pi {\rm{.}}$$

(vi) cot x (variation in ‘b/a’)

$${\text{Domain}} = \mathbb{R} - \left\{ {n\pi } \right\},n \in \mathbb{Z}$$

$${\text{Range}} = \mathbb{R}$$

(x) = cot x is periodic with period $$\pi$$

Can you notice any relation between the graphs of reciprocal pairs?

$\sin x \leftrightarrow {\rm{cosec}}\;x, \qquad \cos x \leftrightarrow {\rm{sec}}\;x, \qquad \tan x \leftrightarrow \cot \;x$

If you can, there should be no problem in drawing the graph of say, cosec x from sin x, or, sec x from cos x etc.

In general, you will learn over time that plotting the graph of \begin{align}\frac{1}{{f\left( x \right)}}\end{align} from the graph of $$f\left( x \right)$$ is quite straightforward. You just have to kept the behavior of the reciprocal function in mind. And what is that behavior?

{\rm{If}}\;x > 0,\;{\rm{then}}\frac{1}{x} > 0: \left\{ {\;\begin{align}{{\rm{If}}\;x \to \infty ,}{\frac{1}{x} \to 0}\\{{\rm{If}}\;x \to 0,} {\frac{1}{x} \to \infty }\end{align}\;} \right\}

{\rm{If}}\;x < 0,\;{\rm{then}}\frac{1}{x} < 0: \left\{ {\begin{align}{\;{\rm{If}}\;x \to - \infty ,}{\frac{1}{x} \to 0}\\{{\rm{If}}\;x \to 0,} {\frac{1}{x} \to - \infty }\end{align}\;} \right\}

In case of any confusion, look at the graph of \begin{align}y = \frac{1}{x}\end{align}again.

12. Polynomials and polynomial ratios

Polynomial functions are function of the form $$f\left( x \right) = {a_0}{x^n} + {a_1}{x^{n - 1}} + ...... + {a_{n.}}$$ $$\left\{ \begin{gathered} n \in {\mathbb{Z}^ + } \hfill \\ {a_i} \in \mathbb{R} \hfill \\ \end{gathered} \right\}$$

They are continuous function (without any breaks in the graph; we will discuss the continuity of functions later) whose domain is $$\mathbb{R}$$ and range is $$\mathbb{R}$$ or a subset of $$\mathbb{R}$$.

For example               $$f\left( x \right) = {x^2} + 2x + 3\,$$ is a quadratic function

$$f\left( x \right) = a{x^3} + b{x^2} + cx + d$$is a cubic function.

Polynomial ratios are functions of the form \begin{align}h(x) = \frac{{f(x)}}{{g(x)}}\end{align}where f (x) and g(x) are themselves polynomials.

The domain will exclude all points where g(x) can become 0 (the roots of g(x)). The range will be a subset of  $$\mathbb{R}$$ (or $$\mathbb{R}$$). The continuity of such functions will be discussed later.

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