Educational

Find Values of x for Equal Functions Find the Set of Values of \( x \) Question: Find the set of values of \(x\) for which the functions \[ f(x)=3x^2-1 \] and \[ g(x)=x+3 \] are equal. Solution: For equal functions, \[ f(x)=g(x) \] Therefore, \[ 3x^2-1=x+3 \] \[ 3x^2-x-4=0 \] Factorizing, \[ 3x^2-4x+3x-4=0 \] […]

Domain of f + g Find the Domain of \( f+g \) Question: Let \(f\) and \(g\) be two functions given by \[ f=\{(2,4),(5,6),(8,-1),(10,-3)\} \] and \[ g=\{(2,5),(7,1),(8,4),(10,13),(11,-5)\} \] Find the domain of \[ f+g \] Solution: Domain of \(f\): \[ \{2,5,8,10\} \] Domain of \(g\): \[ \{2,7,8,10,11\} \] Domain of \(f+g\) is the common

Number of Functions from A to B Find the Number of Functions from \(A\) to \(B\) Question: Let \(A\) and \(B\) be two sets such that \[ n(A)=p \] and \[ n(B)=q. \] Write the number of functions from \(A\) to \(B\). Solution: Each element of set \(A\) can be mapped to any one of

Domain and Range of √([x]-x) Find the Domain and Range of the Function Question: Write the domain and range of the function \[ f(x)=\sqrt{[x]-x} \] Solution: Since \[ [x]\le x

Domain and Range of √(x-[x]) Find the Domain and Range of the Function Question: Write the domain and range of \[ f(x)=\sqrt{x-[x]} \] Solution: Since \[ 0\le x-[x]

Domain and Range of 1/√(x−|x|) Find the Domain and Range of the Function Question: Write the domain and range of the function \[ f(x)=\frac1{\sqrt{x-|x|}} \] Solution: Since square root is in denominator, \[ x-|x|>0 \] Case I: \(x\ge0\) \[ |x|=x \] \[ x-|x|=0 \] Not allowed. Case II: \(x

Find Composite Function Value Find the Value of Composite Function Question: If \(f,\ g,\ h\) are real functions given by \[ f(x)=x^2, \qquad g(x)=\tan x, \qquad h(x)=\log_e x \] then write the value of \[ (h\circ g\circ f)\left(\frac{\sqrt{\pi}}{4}\right) \] Solution: First, \[ f\left(\frac{\sqrt{\pi}}{4}\right) = \left(\frac{\sqrt{\pi}}{4}\right)^2 \] \[ =\frac{\pi}{16} \] Now, \[ g\left(\frac{\pi}{16}\right) = \tan\frac{\pi}{16} \]

Find f(a+1)-f(a-1) Find \( f(a+1)-f(a-1) \) Question: If \[ f(x)=4x-x^2,\qquad x\in R \] then write the value of \[ f(a+1)-f(a-1) \] Solution: \[ f(a+1) = 4(a+1)-(a+1)^2 \] \[ =4a+4-a^2-2a-1 \] \[ =-a^2+2a+3 \] Also, \[ f(a-1) = 4(a-1)-(a-1)^2 \] \[ =4a-4-a^2+2a-1 \] \[ =-a^2+6a-5 \] Therefore, \[ f(a+1)-f(a-1) \] \[ =(-a^2+2a+3)-(-a^2+6a-5) \] \[ =8-4a \]

Domain and Range of Rational Function Find the Domain and Range of the Function Question: Write the domain and range of the function \[ f(x)=\frac{x-2}{2-x} \] Solution: \[ f(x)=\frac{x-2}{2-x} \] \[ =\frac{x-2}{-(x-2)} \] \[ =-1, \qquad x\ne2 \] Therefore, Domain: \[ R-\{2\} \] Range: \[ \{-1\} \] Next Question / Full Exercise

Find f(f(1/x)) Find \( f(f(1/x)) \) Question: If \[ f(x)=1-\frac1x, \] then write the value of \[ f\left(f\left(\frac1x\right)\right). \] Solution: First, \[ f\left(\frac1x\right) = 1-\frac{1}{1/x} \] \[ =1-x \] Now, \[ f(f(1/x)) = f(1-x) \] \[ =1-\frac1{1-x} \] \[ =\frac{1-x-1}{1-x} \] \[ =\frac{x}{x-1} \] Therefore, \[ \boxed{f(f(1/x))=\frac{x}{x-1}} \] Next Question / Full Exercise