Algebra of Real Functions

$f\left(x\right)=x^2+4x+6$ and $g\left(x\right)=x^2+2x+8$. Prove that addition of functions is commutative using given functions.

$f\left(x\right)=2x^2+4x+5$ and $g\left(x\right)=x^2+3x+4$. Prove that addition of functions is commutative using given functions.

$f\left(x\right)=x^2+2x+2$ and $g\left(x\right)=x+1$. Prove that addition of functions is commutative using given functions.

$f\left(x\right)=x^2+4x+2$ and $g\left(x\right)=x-1$. Prove that addition of functions is commutative using given functions.

$f\left(x\right)=x^2+3$ and $g\left(x\right)=x+4$. Prove that addition of functions is commutative using given functions.

If $f\left(x\right)=x^2$ and $g\left(x\right)=2x-1$, then find the value of $\left(f-g\right)\left(41\right)$.

If $f\left(x\right)=x^2$ and $g\left(x\right)=2x+1$, then find the value of $\left(f+g\right)\left(59\right)$.

Reduction of the proper fraction $\frac{f\left(x\right)}{g\left(x\right)}$ into a sum of partial fraction depends on the factorization of

Find the value of $\mathrm{x}$ for which the functions $\mathrm{f}\left(\mathrm{x}\right)=3{\mathrm{x}}^2-1$ and $\mathrm{g}\left(\mathrm{x}\right)=3+\mathrm{x}$ are equal?

If $\mathrm{f}$ and $\mathrm{g}$ are two real valued functions defined as $\mathrm{f}\left(\mathrm{x}\right)=2\mathrm{x}+1, \mathrm{g}\left(\mathrm{x}\right)={\mathrm{x}}^2+1$ then $\frac{\mathrm{f}}{\mathrm{g}}$$=\frac{mx+1}{x^2+1}$. Find $m$.

If $\mathrm{f}$ and $\mathrm{g}$ are two real valued functions defined as $\mathrm{f}\left(\mathrm{x}\right)=2\mathrm{x}+1, \mathrm{g}\left(\mathrm{x}\right)={\mathrm{x}}^2+1$ then find the sum of coefficients of $x^3$ and $x$ in $\mathrm{fg}$

If $\mathrm{f}$ and $\mathrm{g}$ are two real valued functions defined as $\mathrm{f}\left(\mathrm{x}\right)=2\mathrm{x}+1, \mathrm{g}\left(\mathrm{x}\right)={\mathrm{x}}^2+1$ then find the coefficient of $x$ in $\mathrm{f}-\mathrm{g}$

If $\mathrm{f}$ and $\mathrm{g}$ are two real valued functions defined as $\mathrm{f}\left(\mathrm{x}\right)=2\mathrm{x}+1, \mathrm{g}\left(\mathrm{x}\right)={\mathrm{x}}^2+1$,then $\mathrm{f}+\mathrm{g}$$=x^2+kx+2$. Find $k$.

Let $\mathrm{f}$ and $\mathrm{g}$ be real functions defined by $\mathrm{f}\left(\mathrm{x}\right)=2\mathrm{x}+1$ and $\mathrm{g}\left(\mathrm{x}\right)=4\mathrm{x}-7$

For what real number, $\mathrm{f}\left(\mathrm{x}\right)<\mathrm{g}\left(\mathrm{x}\right)$?

If $\mathrm{A}=\{\mathrm{x}:\mathrm{x}\in \mathrm{W},\mathrm{x}<2\} \mathrm{B}=\{\mathrm{x}:\mathrm{x}\in \mathrm{N},1<\mathrm{x}<5\}\mathrm{C}=\{3,5\}$ find

$\mathrm{A}\times (\mathrm{B}\cup \mathrm{C})$

Let $f\left(x\right)$ be a non-zero function satisfying $f\left(x+y\right)=f\left(x\right) f\left(y\right),$ $\forall \mathrm{ }x,\mathrm{ }y\in R$ .Then $f\left(0\right)$ is equal to

Let $f\left(x\right)=a{x}^2+bx+c \text{and} g\left(x\right)=p{x}^2+qx$ with $g\left(1\right)=f\left(1\right), g\left(2\right)-f\left(2\right)=1, g\left(3\right)-f\left(3\right)=4,$ then $g\left(4\right)-f\left(4\right)$is: