Some Important Properties Of Functions

greatest integer and fractional part

Greatest Integer

Any real number x can always think of lying between two consecutive integers say P and P+1. i.e. P $⩽$ x < (P + 1). That means, there always exist an integer, say 'P' which is just less than or equal to x. This unique 'P' is called the greatest integral value of x and is symbolically denoted as [x] i.e. [x] stands for the greatest integer that is less than or equal to x.

e.g. x = 3.54 $\Rightarrow$ 3 < x < 4 $\Rightarrow$ [x] = 3, x = -2.95 $\Rightarrow$ -3 < x < -2 $\Rightarrow$ [x] = - 3

It is obvious that if x is integer, then [x] = x.

Domain $\to$ R; Range $\to$ I;

Period $\to$ non periodic; Nature $\to$ neither even nor odd

Important Points:

$\bullet$ P $⩽$ x < P + 1 $\iff$ [x] = P

$\bullet$ P₁ $⩽$ [x] $⩽$ P₂ $\Rightarrow$ P₁ $⩽$ x < P₂ + 1 $\Rightarrow$ x $\in$ [P₁, P₂ + 1)

e.g. -2 $⩽$ [x] $⩽$2 $\Rightarrow$ -2 $⩽$ x < 3 $\Rightarrow$ x $\in$ [-2, 3)

$\bullet$ x-1 < [x] $⩽$ x

$\bullet$[[x]] = [x]

$\bullet$[n + x] = n + [x] where n $\in$I

$\bullet$ [{\rm{x}}]\,\, + \,\,[ - {\rm{x}}]\,\, = \,\,\left\{ {\begin{array}{*{20}{c}}{\,\,\,\,0\,\,\,\,\,\,if\,{\rm{x}}\, \in \,\,\,{\rm{integer}}.}\\{ - 1\,\,\,\,\,\,\,if\,{\rm{x}} \notin {\rm{integer}}}\end{array}} \right.

Fractional Part:

Fractional Part of any real number is defined as the difference between the number 'x' and it's integral value '[x]' and is symbolically denoted as {x}.

Thus, {x} = x – [x], e.g. if x = 5.68, then [x] = 5 and {x} = 0.68.

If x is an integer $\Rightarrow$ x = [x] $\Rightarrow$ {x} = 0 $\Rightarrow$ {[x]} = 0

If $\text{x}\in \left[0,1\right)$, then [x]=0$\Rightarrow${x}=x

$\text{x}\in \left[1,2\right)$, then [x]=1$\Rightarrow${x}=x-1

Domain $\to$ R; Range $\to$ [0,1);

Period $\to$ 1; Nature $\to$ neither even nor odd

Key Points:

$\bullet$ 0 $⩽$ {x} < 1

$\bullet$ [{x}] = 0, {[x]} = 0

$\bullet$ x – 1 < [x] $⩽$ x, 0 $⩽$ {x} < 1

$\bullet$ {x} + {–x} = $\{\begin{array}{c}0,ifx\in \text{integer}\\ 1,ifx\notin \text{integer}\end{array}$

Illustration 1 Solve for x, $[\text{x}]+\frac{1}{\text{x}-2.7}=2+\frac{1}{\text{x}-2.7}$

$\bullet$ Common Mistake: Student used to solve in this manner

$\bullet$First they cancel the common term both the sides.$\Rightarrow$[x]=2$\Rightarrow \text{x}\in \left[2,3\right)$.

Solution: First step is correct. Cancel common term. But

The important point is that $\frac{1}{\text{x}-2.7}$is not defined when x=2.7, hence exclude this from the solution set.

EVEN AND ODD FUNCTIONS:

If f: X$\to$Y is a real valued function such that for all x $\in$ D $\Rightarrow$ – x$\in$D (where D = domain of f) and if f(– x) = f(x) for every x $\in$ D then f is said to be an even function and if f(– x) = – f(x) then f is said to be odd function.

IMPORTANT POINTS:

$\bullet$Even functions are symmetric about the y – axis (i.e. if (x, y) lies on the curve, then (– x, y) also lies on the curve).

$\bullet$Odd functions are symmetric about the origin and it is placed either in the first and third quadrant or in the second and fourth quadrant. (i.e. if (x, y) lies on the curve, then (– x, – y) also lies on the curve).

$\bullet$f(x) = 0 is the only function which is both even and odd.

$\bullet$If f(x) is an odd function , then f(x) is an odd function provided f(x) is differentiable on R.

$\bullet$To express a given function f(x) as the sum of an even and odd function, we write $f(x)=\left(\frac{f(x)+f(-x)}{2}\right)+\left(\frac{f(x)-f(-x)}{2}\right)$, where $\left(\frac{f(x)+f(-x)}{2}\right)$is an even function and $\left(\frac{f(x)+f(-x)}{2}\right)$is an odd function.

$\bullet$If x = 0 $\in$ domain of f, then for odd function f(x), f(0) = 0 i.e. if for a function, f(0) $\ne$ 0, then that function can not be odd.

Illustration 2 Which of the following functions is (are) even, odd or neither :

(i). f(x) = $\frac{\left|x^2-1\right|\left(cosx\right)}{1+log\left(e^x+e^{-x}\right)}$

(ii). $f\left(x\right)=log\left(\frac{1-x}{1+x}\right)$

(iii). $f\left(x\right)=log\left(x+\sqrt{1+x^2}\right)$

(iv). f (x) = sinx – cosx

Solution: (i) f(x) = $\frac{\left|x^2-1\right|\left(\text{cosx}\right)}{1+log\left(e^x+e^{-x}\right)}$

f(– x) = $\frac{\left|{\left(-x\right)}^2-1\right|\left(\cos \left(-x\right)\right)}{1+log\left(e^{-x}+e^{-\left(-x\right)}\right)}$=$\frac{\left|x^2-1\right|\left(\cos x\right)}{1+log\left(e^x+e^{-x}\right)}$ = f(– x) $\Rightarrow$ f(x) is even.

(ii) $f(-x)=\log \left(\frac{1-(-x)}{1+(-x)}\right)=\log \left(\frac{1+x}{1-x}\right)$= – f(x).Hence f(x) is odd.

(iii) f(– x) = $log\left(-x+\sqrt{1+{(-x)}^2}\right)$

= $log\left(\frac{(-x+\sqrt{1+x^2)}(x+\sqrt{1+x^2})}{(x+\sqrt{1+x^2)}}\right)$ = $log\left(\frac{1}{x+\sqrt{1+x^2}}\right)$ = – f(x).

Hence f(x) is odd.

(iv) f(– x) = sin(– x) – cos(– x) = – sinx – cos x

Hence f(x) is neither even nor odd.

PERIODIC FUNCTION

A function f: X$\to$Y is said to be a periodic function provided there exists a positive real number T such that f(x + T) = f(x), for all x $\in$ X. The least of all such positive numbers T is called the principal period or fundamental period or simply period of f.

$\bullet$ To check the periodicity of a function put f(T+x)=f(x) and solve this equation to find the positive values of t independent of x. If positive values of T independent of x are obtained, then f(x) is a periodic function and the least positive value of T is the period of the function f(x). If no positive value of T independent of x is obtained then f(x) is non-periodic function.

$\bullet$ A constant function is periodic but does not have a well-defined period.

$\bullet$If f(x) is periodic with period p, then f(ax + b) where a, b $\in$ R (a $\ne$ 0) is also period with period p/|a|.

$\bullet$ If f(x) is periodic with period p, then a f(x) + b where a, b $\in$ R (a $\ne$ 0) is also periodic with period p.

$\bullet$ If f(x) is periodic with period p, then f (ax + b) where a, b $\in$ R (a $\ne$ 0) is also period with period p/|a|

$\bullet$ Let f(x) has period p = m/n (m, n $\in$ N and co-prime) and g(x) has period q = r/s (r, s$\in$N and co-prime) and let t be the LCM of p and q i.e. $\text{t = }\frac{\text{LCM of (m, r)}}{\text{HCM of (r, s)}}$, then t shall be the period of f + g provided there does not exist a positive number) k (< t) for which f(k + x) + g(k + x) = f(x)+ g(x), else k will be the period. The same rule is c applicable for any other algebraic combination of f(x) and g(x).

SOME KEY POINTS:

$\bullet$ LCM of p and q always exist if p/q is a rational quantity. If p/q is irrational then algebraic) combination of f and g is non-periodic.

$\bullet$sinⁿx, cosⁿx, cosecⁿx and Secⁿx have period 2$\pi$ if n is odd and p if n is even.

$\bullet$tanⁿx and cotⁿx have period $\pi$ whether n is odd or even.

$\bullet$If g is periodic then fog will always be a periodic function. Period of fog may or may not be the period of g.

$\bullet$ If f is periodic and g is strictly monotonic (other than linear) then fog is non-periodic.

There are two types of questions asked in the examination. You may be asked to test for periodicity of the function or to find the period of the function. In the former case you just need to show that f(x + T) = f(x) for same T (>0) independent of x whereas in the latter, you are required to find a least positive number T independent of x for which f(x +T)= f(x) is satisfied

Illustration 3 Find the period of function sin4x + tan2x.

Solution: Period of sin4x is $\frac{\pi }{2}$, also period of tan 2x is $\frac{\pi }{2}$.

Hence period of f(x) is $\frac{\pi }{2}$