Composition of Functions

Let $f$ and $g$ be real functions defined by $f\left(x\right)=\frac{x}{x+1}$ and $g\left(x\right)=\frac{1}{x+3}$.

Describe the functions $g\circ f$ and $f\circ g$ (if they exist.)

The function $f:R\to R$ is defined by$f\left(x\right)=\sin x+\cos x$ is

If $f$ and $g$ are defined as given below then find $\left(\mathit{g}\mathit{o}\mathit{f}\right)\left(\mathit{x}\right)$:

$f:R\to R, f\left(x\right)=2x+x^{-2}$,$g:R\to R, g\left(x\right)=x^4 +2x+4$.

If  $f:R\to R , f\left(x\right)=x^2+3x+1$ and $g:R\to R, g\left(x\right)=2x-3$, then find $\left(g\mathit{\circ }g\right)\left(x\right)$

If $A=\left\{1, 2, 3, 4\right\}$, $f:R\to R, f\left(x\right)=x^2+3x+1$ and $g:R\to R, g\left(x\right)=2x-3$, then $\left(fof\right)\left(x\right)=a{x}^4+b{x}^3+c{x}^2+dx+5$ then find $a+b+c+d$.

If $A=\left\{1,2,3,4\right\}$, $f:R\to R, f\left(x\right)=x^2+3x+1$ and  $g:R\to R , g\left(x\right)=2x-3$ then $\left(g\mathit{\circ }f\right)\left(x\right)=2x^2+6x-k$. Find the value of $k$.

If $A=\left\{1,2,3,4\right\}$, $f:R\to R, f\left(x\right)=x^2+3x+1$ and $g:R\to R, g\left(x\right)=2x-3$, then $\left(fog\right)\left(x\right)=k_1{x}^2-k_{2}x+1$. Find the value of $k_1+k_2$?

If $f$ and $g$ are defined as $f:R\to R, f\left(x\right)=2x+x^{-2}$,$g:R\to R, g\left(x\right)=x^4 +2x+4$ then $\left(g\circ f\right)\left(x\right)={\left(2x+x^{-2}\right)}^4+a\left(2x+x^{-2}\right)+b$. Find the value of $a+b$?

If $f\left(x\right)=\frac{x–3}{x+1}$, then the value of $f\left[f\left\{f\left(x\right)\right\}\right]$ is equal to

Four functions are defined on set $R_o$, Such that,$f_1\left(x\right)=x, f_2\left(x\right)=-x, f_3\left(x\right)= \frac{1}{x}, f_4\left(x\right)=–\frac{1}{x}$

Construct the composition table for the composition of functions$f_1, f_2, f_3, f_4$. Also, find identity element and inverse of every element.

If three functions $f\mathit{,}\mathit{ }g\mathit{,}\mathit{ }h$ defined from $\mathrm{R}$ to $\mathrm{R}$ in such a way that $f\left(x\right)=x^2, g\left(x\right)=\cos x$ and $h\left(x\right)=2x+3$ 

then find the value of $\left\{h\circ \left(g\circ f\right)\right\}\left(\sqrt{2\pi }\right)$.

If  $f\mathit{:}R\mathit{\to }R\mathit{ }$ and $g\mathit{:}R\mathit{\to }R$ are the two functions such that $f\left(x\right)=3x+4$ and $g\left(x\right)=\frac{1}{3}\left(x-4\right)$

find $\left(\mathit{f}\mathit{o}\mathit{g}\right)\left(\mathit{x}\right)$ and $\left(\mathit{g}\mathit{o}\mathit{f}\right)\left(\mathit{x}\right)$  also find $\left(gog\right)\left(1\right)$

If $f\mathit{:}{R}^{\mathit{+}}\mathit{\to }{R}^{\mathit{+}}$ and $g\mathit{:}{R}^{\mathit{+}}\mathit{\to }{R}^{\mathit{+}}$ are the two functions defined below then $f\left(x\right)=x^2$ and $g\mathit{\left(}x\mathit{\right)}\mathit{=}\sqrt{x}$

find $\left(\mathit{f}\mathit{o}\mathit{g}\right)$ and $\left(\mathit{g}\mathit{o}\mathit{f}\right)$. Are they equal?

If $A\mathit{=}\left\{a\mathit{,}\mathit{ }b\mathit{,}\mathit{ }c\right\}$, $B=\left\{u, v, w\right\}$ and $f:A\to B$and $g:B\to A$ are defined as $f=\left\{ \left(a, v\right), \left(b, u\right) ,\left(c ,w\right)\right\}$ ; $g= \left\{\left(u, b\right) ,\left(v ,a\right), \left(w,c\right)\right\}$
then find $\left(fog\right)$ and $\left(gof\right)$.

If  $f\mathit{:}R\mathit{\to }R\mathit{ }$and $g\mathit{:}R\mathit{\to }R$ are the two functions defined below then find $\left(\mathit{f}\mathit{o}\mathit{g}\right)\left(\mathit{x}\right)$ and $\left(\mathit{g}\mathit{o}\mathit{f}\right)\left(\mathit{x}\right)$

$f\left(x\right)=x^2+2x+3 , g\left(x\right)=3x-4$

If  $f\mathit{:}R\mathit{\to }R\mathit{ }$ and $g\mathit{:}R\mathit{\to }R$ are the two functions defined below then find $\left(fog\right)\left(x\right)$ and $\left(gof\right)\left(x\right)$

$f\left(x\right)=x , g\left(x\right)=\left|x\right|$

If  $f:R\to R$ and $g\mathit{:}R\mathit{\to }R$ are the two functions defined below then find $\left(fog\right)\left(x\right)$ and $\left(gof\right)\left(x\right)$

$f\left(x\right)=x^2+8 , g\left(x\right)=3x^3+1$

Let $f\left(x\right)=x^2, g\left(x\right)={\log }_{\mathrm{e}}x.$ The number of values of $x$ for which $\left(fog\right)\left(x\right)=\left(gof\right)\left(x\right)$ is

Let $f\left(x\right)=\frac{e^x-e^{-x}}{2}$ and if $g\left(f\left(x\right)\right)=x$, then $g\left(\frac{e^{1002}-1}{2e^{501}}\right)$ is equal to

If $f\left(x\right)=\frac{x+1}{x-1},$ then the value of $f\left(f\left(x\right)\right)$ is equal to