# Domains and Ranges of Standard Functions Set 3

**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’) **

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

\({\text{Range}} = \mathbb{R}\)

*f *(*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.