Domains and Ranges of Standard Functions Set 2

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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, f (x­­­­­­­­) drops again to 0, and then starts increasing as x increases. And so on, we see that at each integral value of x, f (x) becomes 0, then increases and approaches 1 as 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)\(\left\{ { - x} \right\} = 1 - \left\{ x \right\},n \in \mathbb{Z}\)

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