# Domains and Ranges of Standard Functions Set 2

**6.** **Reciprocal function \(\begin{align}f\,(x) = \frac{1}{x}\end{align}\)** ** **

Domain = \(\mathbb{R}\) – {0}

Range = \(\mathbb{R}\)– {0}

{Range is \(\mathbb{R}\) – {0} because for no value of *x* is *f* (*x*) = 0}

**7.** **Step function \(f\,(x)\, = \,0\;\; if \;x < 0\)**

**\(1\;\; if\; x \geqslant 0 \)**

Domain = \(\mathbb{R}\)

Range = {0,1}

**8.** **Modulus function \(f\left( x \right) = \left| x \right|\, = \,\,\,x\,\,\,\,if\,\,x \ge \;0\)** ** **

** \(- x\,\;\,if\;x\; < \;0\)**

Basically, this function gives the magnitude of a number and strips it off its negative sign,if it is negative.

Domain=\(\mathbb{R}\)

Range = [0, \(\infty\))

**9. Greatest integer function \(f\,(x)=[x]\) **

This is an interesting function. It is defined as the largest integer less than or equal to \(x\).

For example, [1.3] = 1 [–1.3] = –2 [0.6] = 0 [\(\pi\)] =3 etc.

To draw its graph, we note that **\(f\,(x)=0\)** for \(0 \le x < 1\), **\(f\,(x)=1\)** for \(1 \le x < 2\), **\(f\,(x)=-1\) **for \( - 1 \le x < 0\) and so on.

Therefore, if \(x\) lies in the interval** \([n, n + 1)\)**, then the value of **\([x]\) **is** \(n\)**.

Domain = \(\mathbb{R}\)

Range = \(\mathbb{Z}\) (set of all integers)

Properties

(i) \([x + n] = [x] + n,n \in \mathbb{Z}\)

(ii) \([ - x] = - [x] - 1,\;x \notin\mathbb{Z}\)

\([ - x] = - [x],\;\;\;\;\;\;\;\;x \in \mathbb{Z}\)

Verify these properties carefully because they are very important

**10.** ** Fractional part f (x) = {x}**

Every number *x* can be written as the sum of its integer and fractional parts. So, fractional part

**{ x} = x – [x]**

or in other words, fractional part of a number is the difference between that number and its integral part **[ x]**

We example, **{1.3}= 0.3, {-1.3} = 0.7** and so on.

We can obviously see that \(0 \le \{ x\} < 1\)

To draw the graph, we note that ** f (x) = x** for \(0\underline < x < 1\) because in this interval, [

*x*] = 0. As soon as

*x*becomes 1,

**drops again to 0, and then starts increasing as**

*f*(*x*)*x*increases. And so on, we see that at each integral value of

*x*,

**f****(**becomes 0, then increases and approaches 1 as

*x*)*x*approaches the next integer, and finally drops again to 0, never becoming 1.

**Properties**

(i) \(\left\{ {x + n} \right\} = \left\{ x \right\},n \in \mathbb{Z}\)

(ii)\(\left\{ { - x} \right\} = 1 - \left\{ x \right\},n \in \mathbb{Z}\) ** **

\(\left\{ { - x} \right\} = 0,\;\;\;n \in \mathbb{Z}\)