Composition of Functions

If the function $f:R\to R \text{and} g:R\to R$ are given by $f\left(x\right)=\left|x\right|$ and $g\left(x\right)=\left[x\right]. \left(\left[x\right]\text{ is greatest integer}\right).$ Find $fog\left(-\frac{1}{2}\right) \text{and} gof\left(-\frac{1}{2}\right).$

Find the composite functions $\left(f\circ g\right)\left(x\right)$ and $\left(g\circ f\right)\left(x\right)$ for the functions $f\left(x\right)=x^2-3$ and $g\left(x\right)=\sqrt{x+1}+5$.

Let $x_1$be a positive real number and for every integer $n\ge 1$ let $x^{n+1}=1+x_1{x}_2{x}_3....x_n$ . If $x_5=43$, what is the sum of the digits of the largest prime factor of $x_6?$

If $f\left(x\right)=-x^2+x+m, g\left(x\right)=x+3$ and $f\left(g\left(f\left(2\right)\right)\right)=-1$. Then what is the value of $m$?

If $f\left(x\right)=\sqrt{4-x^2},g\left(x\right)=x^2$ then find $g\left(f\left(x\right)\right)$.

If $f:R\to R$ and $g:R\to R$ are defined by $f\left(x\right)=\left|x\right|$ and $g\left(x\right)=\left[x-3\right]$ for $x\in R,$ then $\left\{g\left(f\left(x\right)\right):\frac{-8}{5}<x<\frac{8}{5}\right\}=?$

Let $f\left(x\right)$ and $g\left(x\right)$ be two real polynomials of degree $2$ and $1$ respectively. If $f\left(g\left(x\right)\right)=8x^2-2x,$ and $g\left(f\left(x\right)\right)=4x^2+6x+1,$ then find the value of $f\left(2\right)+g\left(2\right).$ 

If $f\left(x\right)=x^2+x+7 \& g\left(x\right)=5x-3$ where $f\left(x\right) \& g\left(x\right)$ are real functions find $fog\left(2\right) \& gof\left(1\right)$.

Find $fog$ and $gof$ if $f\left(x\right)=3x^2-5x+2, g\left(x\right)=\sqrt{x}$.

If $f\left(x\right)=x^2$ then $fofofof\left(x\right)=$

If $f\left(x\right)=2x+1 \text{and} g\left(x\right)=x^2-2$ then find $\left(gof\right)\left(x\right)$

If $f\left(x\right)=\sin x$ and $g\left(x\right)=\left[x\right]$, where $\left[\right]$ denotes the greatest integer function, be two real functions, then find $fog\left(\frac{-\mathrm{\pi }}{2}\right)$.

Consider $f:N\to N, g:N\to N$ and $h:N\to R$ defined as $f\left(x\right)=2x, g\left(y\right)=3y+4$ and $h\left(z\right)=\sin \mathit{ }z,\forall x, y$ and $z$ in $N$. Show that $ho\left(gof\right)=\left(hog\right) of$.

Let $f:\left\{1, 2, 3\right\}\to \left\{a, b, c\right\}$ be one-one and onto function given by $f\left(1\right)=a, f\left(2\right)=b$ and $f\left(3\right)=c$. Show that there exists a function $g:\left\{a, b, c\right\}\to \left\{1, 2, 3\right\}$ such that $gof=I_{\mathit{X}}$ and $fog=I_{\mathit{Y}}$, where, $X=\left\{1, 2, 3\right\}$ and $Y=\left\{a, b, c\right\}$.

Are $f$ and $g$ both necessarily onto, if $gof$ is onto?

Consider functions $f$ and $g$ such that composite $g$ of is defined and is one-one. Are $f$ and $g$ both necessarily one-one.

Show that if $f:A\to B$ and $g:B\to C$ are onto, then $gof:A\to C$ is also onto.

Show that if $f:A\to B$ and $g:B\to C$ are one-one, then $gof:A\to C$ is also one-one.

Show that if $f:R-\left\{\frac{7}{5}\right\}\to R-\left\{\frac{3}{5}\right\}$ is defined by $f\left(x\right)=\frac{3x+4}{5x-7}$ and $g:R-\left\{\frac{3}{5}\right\}\to R-\left\{\frac{7}{5}\right\}$ is defined by $g\left(x\right)=\frac{7x+4}{5x-3}$, then $fog=I_{\mathit{A}}$ and $gof=I_{\mathit{B}}$, where, $A=R-\left\{\frac{3}{5}\right\}, B=R-\left\{\frac{7}{5}\right\};\mathit{ }{I}_A\left(x\right)=x,\forall x\in A, I_B\left(x\right)=x,\forall x\in B$ are called identity functions on sets $A$ and $B$, respectively.

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