Composition of Functions

If $f:R^{+}\to R^{+}$and $g:R^{+}\to R^{+}$, defined as below

$f\left(x\right) = x^2$, $g\left(x\right) = \sqrt{x},$ then $gof$ and $fog$, are equal functions.

If $f :R\to R$ and $g :R\to R$, such that $f\left(x\right)=3x+4$ and $g\left(x\right) = \frac{(x –4)}{3}$ then, find the value of $\left(gog\right)\left(1\right)$.

The composition of functions is not commutative.

If $f:R\to R$ is defined by $f\left(x\right)=x^2-3x+2$, write $f\left(f\right(x\left)\right)$.

Let $f:R\to R$ and $g:R\to R$ be defined by $f\left(x\right)=2x+1$ and $g\left(x\right)=x^2-2$ respectively. If $\left(gof\right)\left(x\right)=a{x}^2+ax-1$, find the value of $a$.

Find $k,$ if $f\left(k\right)=2k-1$ and $f\circ f\left(k\right)=5$

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

If $f:R\to R$ and $g:R\to R$ are defined such that $f\left(x\right)=x^2+3; g\left(x\right)=1-\frac{1}{(1-x)}$ then find $gof\left(x\right)$ and $fog\left(x\right)$.

If $f:R\to R$ and $g:R\to R$ are given by $f\left(x\right)=\cos x$ and $g\left(x\right)=3x^2$. Find $gof$ and $fog$.

If $f\left(x\right)=\frac{x-3}{x+1}$, then $f\left[f\right\{f\left(x\right)\left\}\right]$ equals to

If $f:R\to R, f\left(x\right)=\sin x$ and $g:R\to R, g\left(x\right)=x^2,$ then $g\circ f\left(x\right)=?$

If $f:R\to R, g:R\to R$ defined by $f\left(x\right)=x^2-3x+4$ and $g\left(x\right)=2x+1$, then the value of $x$ for which $f\left(x\right)=\mathit{fog}\left(x\right)$ is

If $f\left(x\right)=e^x$ and $g\left(x\right)=ln\left(x\right)$ for all $x\in [1, \infty )$ then $fog:[1,\infty )\to [1,\infty )$ is

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

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

Define $fog\left(x\right)$ and $gof\left(x\right)$. Also find their domain and range. $f\left(x\right)=\left[x\right], g\left(x\right)=\sin x$.

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 $a+b$?