Find f(x+y)f(x−y) Find \( f(x+y)f(x-y) \) Question: If \[ f(x)=\frac{2^x+2^{-x}}{2} \] then \[ f(x+y)f(x-y) \] is equal to (a) \(\frac12\{f(2x)+f(2y)\}\) (b) \(\frac12\{f(2x)-f(2y)\}\) (c) \(\frac14\{f(2x)+f(2y)\}\) (d) \(\frac14\{f(2x)-f(2y)\}\) Solution: \[ f(x+y)=\frac{2^{x+y}+2^{-(x+y)}}{2} \] \[ f(x-y)=\frac{2^{x-y}+2^{-(x-y)}}{2} \] Multiplying, \[ f(x+y)f(x-y) \] \[ = \frac14 \left(2^{2x}+2^{-2x}+2^{2y}+2^{-2y}\right) \] \[ = \frac12 \left[ \frac{2^{2x}+2^{-2x}}{2} + \frac{2^{2y}+2^{-2y}}{2} \right] \] \[ = \frac12\{f(2x)+f(2y)\} \] […]

Find the Value of the Given Expression Find the Value of the Given Expression Question: If \( f(x)=\cos(\log x) \), then find the value of \[ f(x)f(4)-\frac12\left\{ f\left(\frac{x}{4}\right)+f(4x) \right\} \] (a) \(1\) (b) \(-1\) (c) \(0\) (d) \(\pm1\) Solution: Using \[ 2\cos A\cos B=\cos(A-B)+\cos(A+B) \] \[ 2f(x)f(4) = 2\cos(\log x)\cos(\log 4) \] \[ = \cos(\log

Find f(2x/(1+x²)) Find \( f\left(\frac{2x}{1+x^2}\right) \) Question: If \[ f(x)=\log\left(\frac{1+x}{1-x}\right) \] then \[ f\left(\frac{2x}{1+x^2}\right) \] is equal to (a) \(\{f(x)\}^2\) (b) \(\{f(x)\}^3\) (c) \(2f(x)\) (d) \(3f(x)\) Solution: \[ f\left(\frac{2x}{1+x^2}\right) = \log\left( \frac{1+\frac{2x}{1+x^2}} {1-\frac{2x}{1+x^2}} \right) \] \[ = \log\left( \frac{(1+x)^2}{(1-x)^2} \right) \] \[ = 2\log\left( \frac{1+x}{1-x} \right) \] \[ =2f(x) \] \[ \boxed{\text{Correct Answer: (c)}} \]

Number of Functions from A into B Number of Functions from A into B Question: If \( A=\{1,2,3\} \), \( B=\{x,y\} \), then the number of functions that can be defined from \(A\) into \(B\) is (a) \(12\) (b) \(8\) (c) \(6\) (d) \(3\) Solution: Number of functions from a set having \(m\) elements to

Find f(g(x)) Find \( f(g(x)) \) Question: If \[ f(x)=\log\left(\frac{1+x}{1-x}\right) \] and \[ g(x)=\frac{3x+x^3}{1+3x^2} \] then \( f(g(x)) \) is equal to (a) \(f(3x)\) (b) \(\{f(x)\}^3\) (c) \(3f(x)\) (d) \(-f(x)\) Solution: \[ f(g(x)) = \log\left( \frac{1+g(x)}{1-g(x)} \right) \] Put \[ g(x)=\frac{3x+x^3}{1+3x^2} \] Then, \[ \frac{1+g(x)}{1-g(x)} = \frac{1+\frac{3x+x^3}{1+3x^2}} {1-\frac{3x+x^3}{1+3x^2}} \] \[ = \frac{(1+x)^3}{(1-x)^3} \] Therefore, \[

Which of the Following are Functions? Which of the Following are Functions? Question: Which of the following are functions? (a) \( \{(x,y): y^2=x,\; x,y\in R\} \) (b) \( \{(x,y): y=|x|,\; x,y\in R\} \) (c) \( \{(x,y): x^2+y^2=1,\; x,y\in R\} \) (d) \( \{(x,y): x^2-y^2=1,\; x,y\in R\} \) Solution: A relation is a function if each

Range of f(x)=cos[x] Find the Range of \( f(x)=\cos[x] \) Question: The range of \[ f(x)=\cos[x] \] for \[ -\frac{\pi}{2}

Let f(x)=|x−1| Then Which Statement is Correct? Let \( f(x)=|x-1| \), Then Which Statement is Correct? Question: Let \( f(x)=|x-1| \). Then, (a) \( f(x^2)=[f(x)]^2 \) (b) \( f(x+y)=f(x)f(y) \) (c) \( f(|x|)=|f(x)| \) (d) none of these Solution: Check option (c): \[ f(|x|)=||x|-1| \] and \[ |f(x)|=||x-1|| \] These are not equal in general.

Find the Value of f(x)f(y) Find the Value of the Given Expression Question: If \( f(x)=\cos(\log x) \), then find the value of \[ f(x)f(y)-\frac12\left\{ f\left(\frac{x}{y}\right)+f(xy) \right\} \] (a) \(-1\) (b) \(\frac12\) (c) \(-2\) (d) none of these Solution: \[ f(x)=\cos(\log x), \qquad f(y)=\cos(\log y) \] Using \[ 2\cos A\cos B=\cos(A-B)+\cos(A+B) \] \[ f(x)f(y) =

Find the Value of f(x²)f(y²) Find the Value of the Given Expression Question: If \( f(x)=\cos(\log x) \), then find the value of \[ f(x^2)f(y^2)-\frac12\left\{f\left(\frac{x^2}{y^2}\right)+f(x^2y^2)\right\} \] (a) \(-2\) (b) \(-1\) (c) \(\frac12\) (d) none of these Solution: \[ f(x^2)=\cos(2\log x), \qquad f(y^2)=\cos(2\log y) \] Using \[ 2\cos A\cos B=\cos(A-B)+\cos(A+B) \] \[ f(x^2)f(y^2) = \frac12\left[ f\left(\frac{x^2}{y^2}\right)