Find the Range of f(x)=|x−4|/(x−4) Find the Range of \(f(x)=\dfrac{|x-4|}{x-4}\) Question Find the range of the function \[ f(x)=\frac{|x-4|}{x-4} \] Solution Given \[ f(x)=\frac{|x-4|}{x-4} \] We consider two cases based on the sign of \(x-4\). Case 1: \(x>4\) Then, \[ |x-4|=x-4 \] Therefore, \[ f(x)=\frac{x-4}{x-4}=1 \] Case 2: \(x

Find the Domain of f(x)=1/√([x]²−3[x]+2) Find the Domain of \(f(x)=\dfrac1{\sqrt{[x]^2-3[x]+2}}\) Question Find the domain of the function \[ f(x)=\frac1{\sqrt{[x]^2-3[x]+2}} \] where \([x]\) denotes the greatest integer function. Solution Given \[ f(x)=\frac1{\sqrt{[x]^2-3[x]+2}} \] Since the square root is in the denominator, we require \[ [x]^2-3[x]+2>0 \] Let \[ [x]=t \] Then, \[ t^2-3t+2>0 \] Factorize: \[

Find the Domain and Range of f(x)=(2−x)/(x−2) Find the Domain and Range of \(f(x)=\dfrac{2-x}{x-2}\) Question The domain and range of the function \[ f(x)=\frac{2-x}{x-2} \] are __________ and __________ respectively. Solution Given \[ f(x)=\frac{2-x}{x-2} \] Rewrite the numerator: \[ 2-x=-(x-2) \] Therefore, \[ f(x)=\frac{-(x-2)}{x-2} \] \[ f(x)=-1 \qquad \text{for } x\ne2 \] Domain The denominator

Find the Domain of f(x)=√(9−x)+1/√(x²−16) Find the Domain of \(f(x)=\sqrt{9-x}+\dfrac1{\sqrt{x^2-16}}\) Question Find the domain of the function \[ f(x)=\sqrt{9-x}+\frac1{\sqrt{x^2-16}} \] Solution Given \[ f(x)=\sqrt{9-x}+\frac1{\sqrt{x^2-16}} \] For the function to be defined: The quantity inside each square root must be non-negative. The denominator must not be zero. Condition 1 From \[ \sqrt{9-x} \] we must have

Solve f(x)=0 for f(x)=[x]²−5[x]+6 Solve \(f(x)=0\) for \(f(x)=[x]^2-5[x]+6\) Question If \[ f(x)=[x]^2-5[x]+6 \] then the set of values of \(x\) satisfying \[ f(x)=0 \] is __________. Solution Given \[ f(x)=[x]^2-5[x]+6 \] We need to solve \[ [x]^2-5[x]+6=0 \] Let \[ [x]=t \] Then, \[ t^2-5t+6=0 \] Factorize: \[ (t-2)(t-3)=0 \] Therefore, \[ t=2 \quad \text{or}

Find f(1/x) if f(x)=(x−1)/(x+1) Find \(f\left(\frac1x\right)\) Question If \[ f(x)=\frac{x-1}{x+1} \] then \[ f\left(\frac1x\right) \] is equal to __________. Solution Given \[ f(x)=\frac{x-1}{x+1} \] Replace \(x\) by \(\frac1x\): \[ f\left(\frac1x\right) = \frac{\frac1x-1}{\frac1x+1} \] Simplify numerator and denominator: \[ = \frac{\frac{1-x}{x}}{\frac{1+x}{x}} \] \[ = \frac{1-x}{1+x} \] Multiply numerator and denominator by \(-1\): \[ = -\frac{x-1}{x+1} \]

Find f(1/x)+f(x) if f(x)=(x−1)/(x+1) Find \(f\left(\frac1x\right)+f(x)\) Question If \[ f(x)=\frac{x-1}{x+1} \] then \[ f\left(\frac1x\right)+f(x) \] is equal to __________. Solution Given \[ f(x)=\frac{x-1}{x+1} \] First find \[ f\left(\frac1x\right) \] \[ f\left(\frac1x\right) = \frac{\frac1x-1}{\frac1x+1} \] Simplify numerator and denominator: \[ = \frac{\frac{1-x}{x}}{\frac{1+x}{x}} \] \[ = \frac{1-x}{1+x} \] \[ = -\frac{x-1}{x+1} \] Therefore, \[ f\left(\frac1x\right)=-f(x) \] Hence,

Find the Domain of f(x)=(x²+1)/(x²−3x+2) Find the Domain of \(f(x)=\dfrac{x^2+1}{x^2-3x+2}\) Question Find the domain of the function \[ f(x)=\frac{x^2+1}{x^2-3x+2} \] Solution Given \[ f(x)=\frac{x^2+1}{x^2-3x+2} \] For a rational function, the denominator must not be zero. Therefore, \[ x^2-3x+2\ne0 \] Factorize the denominator: \[ x^2-3x+2=(x-1)(x-2) \] Hence, \[ (x-1)(x-2)\ne0 \] Therefore, \[ x\ne1 \quad \text{and} \quad

Find the Values of x for Which f(x)=g(x) Find the Values of \(x\) for Which \(f(x)=g(x)\) Question The functions \[ f(x)=3x^2-1 \] and \[ g(x)=3+x \] are equal for which values of \(x\)? Solution Given \[ f(x)=3x^2-1 \] and \[ g(x)=3+x \] To find the values of \(x\) for which the functions are equal, equate

Find the Domain of fg for Given Functions Find the Domain of \(fg\) Question Let \(f\) and \(g\) be two real functions given by \[ f=\{(10,1),(2,0),(3,-4),(4,2),(5,1)\} \] and \[ g=\{(1,0),(2,3),(3,-1),(4,4),(5,3)\} \] Then the domain of \(fg\) is given by __________. Solution The domain of the product \(fg\) consists of all elements which belong to both