Educational

Find the Range of f(x)=(x+2)/|x+2| Find the Range of \(f(x)=\dfrac{x+2}{|x+2|}\) Question Find the range of the function \[ f(x)=\frac{x+2}{|x+2|} \] Solution Given \[ f(x)=\frac{x+2}{|x+2|} \] We consider two cases depending on the sign of \(x+2\). Case 1: \(x+2>0\) That is, \[ x>-2 \] Then, \[ |x+2|=x+2 \] Therefore, \[ f(x)=\frac{x+2}{x+2}=1 \] Case 2: \(x+2

Find the Range of f(x)=[x]-x Find the Range of \(f(x)=[x]-x\) Question Find the range of the function \[ f(x)=[x]-x \] where \([x]\) denotes the greatest integer function. Solution Given \[ f(x)=[x]-x \] Let \[ x=n+t \] where \(n\in\mathbb{Z}\) and \[ 0\le t

Find the Domain of f(x)=1/√(|x|-x) Find the Domain of \(f(x)=\dfrac{1}{\sqrt{|x|-x}}\) Question Find the domain of the function \[ f(x)=\frac{1}{\sqrt{|x|-x}} \] Solution Given \[ f(x)=\frac{1}{\sqrt{|x|-x}} \] Since the square root appears in the denominator, the quantity inside the root must be strictly positive. \[ |x|-x>0 \] Case 1: \(x \ge 0\) For \(x \ge 0\), \[

Find f(y) if y = (ax+b)/(cx-d) Find \(f(y)\) if \(y=f(x)=\dfrac{ax+b}{cx-d}\) Question If \[ y=f(x)=\frac{ax+b}{cx-d} \] then find \[ f(y) \] Solution Given \[ y=\frac{ax+b}{cx-d} \] Now, \[ f(y)=\frac{ay+b}{cy-d} \] Substitute the value of \(y\): \[ f(y)= \frac{ a\left(\frac{ax+b}{cx-d}\right)+b }{ c\left(\frac{ax+b}{cx-d}\right)-d } \] Simplify the numerator: \[ = \frac{ \frac{a(ax+b)+b(cx-d)}{cx-d} }{ \frac{c(ax+b)-d(cx-d)}{cx-d} } \] \[ =

Find f(y) if f(x)=x/(x−1)=1/y Find \(f(y)\) if \(f(x)=\frac{x}{x-1}=\frac1y\) Question If \[ f(x)=\frac{x}{x-1}=\frac1y \] then find \[ f(y) \] Solution Given \[ \frac{x}{x-1}=\frac1y \] Cross multiply: \[ xy=x-1 \] Rearranging, \[ x-xy=1 \] \[ x(1-y)=1 \] \[ x=\frac1{1-y} \] Now, \[ f(y)=\frac{y}{y-1} \] From \[ x=\frac1{1-y} \] we get \[ 1-y=\frac1x \] \[ y=1-\frac1x \] Therefore,

Total Number of Functions from Set A to Set B Total Number of Functions from Set A to Set B Question Let \(A\) and \(B\) be any two sets such that \[ n(A)=p \] and \[ n(B)=q \] Then the total number of functions from \(A\) to \(B\) is equal to ? Solution Let \[

Find (f+g)(x), (f-g)(x), (fg)(x) and (f/g)(x) Find \((f+g)(x)\), \((f-g)(x)\), \((fg)(x)\) and \((f/g)(x)\) Question Let \[ f(x)=x^2 \] and \[ g(x)=2x+1 \] be two real functions. Find \((f+g)(x)\), \((f-g)(x)\), \((fg)(x)\) and \((f/g)(x)\). Solution Given \[ f(x)=x^2 \] and \[ g(x)=2x+1 \] Find \((f+g)(x)\) By definition, \[ (f+g)(x)=f(x)+g(x) \] \[ =x^2+(2x+1) \] \[ =x^2+2x+1 \] Therefore, \[

Find f+g, f-g, fg and f/g | Functions Class 11 Maths Find \(f+g\), \(f-g\), \(fg\) and \(f/g\) Question Let \[ f:[0,\infty)\to\mathbb{R} \] and \[ g:\mathbb{R}\to\mathbb{R} \] be defined by \[ f(x)=\sqrt{x} \] \[ g(x)=x \] Find \(f+g\), \(f-g\), \(fg\) and \(\frac{f}{g}\). Solution Given \[ f(x)=\sqrt{x} \] and \[ g(x)=x \] Domains of the Functions Domain

Find f+g, f-g and f/g | Functions Class 11 Maths Find \(f+g\), \(f-g\) and \(f/g\) Question Let \(f, g:\mathbb{R}\to\mathbb{R}\) be defined respectively by \[ f(x)=x+1 \] \[ g(x)=2x-3 \] Find \(f+g\), \(f-g\) and \(\frac{f}{g}\). Solution Given \[ f(x)=x+1 \] and \[ g(x)=2x-3 \] Find \(f+g\) By definition, \[ (f+g)(x)=f(x)+g(x) \] \[ =(x+1)+(2x-3) \] \[ =3x-2

Draw the Graph of Piecewise Function f(x) Draw the Graph of the Function \(f(x)\) Question The function \(f\) is defined by \[ f(x)= \begin{cases} 1-x, & x0 \end{cases} \] Draw the graph of \(f(x)\). Solution The function is defined in three parts. Case 1: \(x0\), the point at \(x=0\) is not included. Therefore, draw an