Educational

Find f⁻¹(9) if f(x)=x² Find \( f^{-1}(9) \) if \( f(x)=x^2 \) Question: If \( f : Q \to Q \) is defined as \( f(x)=x^2 \), then \( f^{-1}(9) \) is equal to (a) \(3\) (b) \(-3\) (c) \(\{-3,3\}\) (d) \(\phi\) Solution: \( f^{-1}(9) \) means the set of all rational numbers whose image […]

Which of the Following is a Function from A to B? Which of the Following is a Function from A to B? Question: Let \( A=\{1,2,3\}, \; B=\{2,3,4\} \), then which of the following is a function from \(A\) to \(B\)? (a) \( \{(1,2),(1,3),(2,3),(3,3)\} \) (b) \( \{(1,3),(2,4)\} \) (c) \( \{(1,3),(2,2),(3,3)\} \) (d) \(

Number of Elements in an Identity Function on a Set of Four Elements Number of Elements in an Identity Function Question The number of elements of an identity function defined on a set containing four elements is __________. Solution Let the set be \[ A=\{a,b,c,d\} \] An identity function on a set maps every element

Find f(3x+2) if f(2x+3)=4x²+12x+15 Find \(f(3x+2)\) Question If \[ f(2x+3)=4x^2+12x+15 \] then find \[ f(3x+2) \] Solution Given \[ f(2x+3)=4x^2+12x+15 \] Let \[ t=2x+3 \] Then, \[ x=\frac{t-3}{2} \] Substitute into the given expression: \[ f(t)=4\left(\frac{t-3}{2}\right)^2 +12\left(\frac{t-3}{2}\right)+15 \] \[ =(t-3)^2+6(t-3)+15 \] \[ =t^2-6t+9+6t-18+15 \] \[ =t^2+6 \] Therefore, \[ f(t)=t^2+6 \] Now replace \(t\) by

Find the Domain of f(x)=(|x|−2)/(|x|−3) Find the Domain of \(f(x)=\dfrac{|x|-2}{|x|-3}\) Question Find the domain of the function \[ f(x)=\frac{|x|-2}{|x|-3} \] Solution Given \[ f(x)=\frac{|x|-2}{|x|-3} \] For the function to be defined, the denominator must not be zero. Therefore, \[ |x|-3\ne0 \] \[ |x|\ne3 \] Hence, \[ x\ne3 \quad \text{and} \quad x\ne-3 \] Thus all real

Find the Domain of f(x)=Σ[1/|2x−n|] Find the Domain of \(f(x)=\displaystyle\sum_{n=1}^{10}\frac1{|2x-n|}\) Question Find the domain of the function \[ f(x)=\sum_{n=1}^{10}\frac1{|2x-n|} \] Solution Given \[ f(x)=\sum_{n=1}^{10}\frac1{|2x-n|} \] Since the denominator cannot be zero, we must have \[ |2x-n|\ne0 \] for every value of \(n=1,2,3,\ldots,10\). Therefore, \[ 2x-n\ne0 \] \[ 2x\ne n \] \[ x\ne\frac n2 \] for

Find the Range of f(x)=√(1−x²) Find the Range of \(f(x)=\sqrt{1-x^2}\) Question Find the range of the function \[ f(x)=\sqrt{1-x^2} \] Solution Given \[ f(x)=\sqrt{1-x^2} \] Step 1: Find the Domain Since the quantity inside the square root must be non-negative, \[ 1-x^2\ge0 \] \[ x^2\le1 \] \[ -1\le x\le1 \] Step 2: Find the Range

Find the Domain of f(x)=x+[x] Find the Domain of \(f(x)=x+[x]\) Question Find the domain of the function \[ f(x)=x+[x] \] where \([x]\) denotes the greatest integer function. Solution Given \[ f(x)=x+[x] \] The function \(x\) is defined for all real numbers. Also, the greatest integer function \([x]\) is defined for every real number \(x\). Therefore,

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: \[