Composite Functions

If $f\left(x\right)=x^2+2$ and $g\left(x\right)=2x-3$, then what is $\left(fg\right)\left(1\right)$ equal to?

For $x\in \mathrm{R},$ two real valued functions $f\left(x\right)$ and $g\left(x\right)$ are such that, $g\left(x\right)=\sqrt{x}+1$ and $fog\left(x\right)=x+3-\sqrt{x}.$ Then $f\left(0\right)$ is equal to 

If $f\left(x\right)=\frac{\left(\tan 1^{°}\right)x+{\log }_e\left(123\right)}{x {\log }_e\left(1234\right)-\left(\tan 1^{°}\right)}, x>0$, then the least value of $f\left(f\left(x\right)\right)+f\left(f\left(\frac{4}{x}\right)\right)$ is

If $f\left(x\right)=\frac{1}{1-x^2}$ and $g\left(x\right)=x$, then $fog\left(x\right)$ is

$f:R\to R, f\left(x\right)=x^2+2x+1 \text{and} g:R\to R.$

If $f\left(g\left(x\right)\right)=x^2+6x+9$ then $g\left(2\right)=$

If $f\left(x\right)=3x+4, x\in R$, $g\left(x\right)=\left\{\begin{array}{l}{x}^2; -1\le x\le 2\\ x^2+2; 2<x\le 3\end{array}\right.$ and $h\left(x\right)=\left\{\begin{array}{l}x+1; x\le 1\\ 2x+1; 1<x\le 2\end{array}\right.$, then function $fogoh\left(\left|x\right|\right)$ is equal to

Let $f,g:R\to R$ be given by $f\left(x\right)=x$ and $g\left(x\right)=2x$, then $fog\left(1\right)$ is

Let $f\left(x\right)=x^2$ and $g\left(x\right)=\sqrt{9+x}$, then the value of $\left(fog-gof\right)\left(4\right)$ is equal to

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 $\mathrm{ℝ}$ be the set of all real numbers. Let $f:\mathrm{ℝ}\to \mathrm{ℝ}$ be a function defined by

$f\left(x\right)=\left\{\begin{array}{ccc}2x-5& \mathrm{if}& x<-3\\ x+2& \mathrm{if}& -3\le x<5\\ 3x+1& \mathrm{if}& x\ge 5\end{array}\right.$

Match the following

The correct match is

If $f\left(f\left(0\right)\right)=0$, where $f\left(x\right)=x^2+ax+b,b\ne 0$ then $a+b=$

If $f\left(x\right)=\left\{\begin{array}{l}1-x, x\le 0\\ x^2, x>0\end{array}\right.$ and $g\left(x\right)=\left\{\begin{array}{l}-x, x<1\\ 1-x, x\ge 1\end{array}\right.$, then $\left(fog\right)\left(x\right)$ is

Let $f:\left[0, 1\right]\to \left[-1, 1\right]$ and $g:\left[-1, 1\right]\to \left[0, 2\right]$ be two functions such that $g$ is injective and $gof:\left[0, 1\right]\to \left[0, 2\right]$ is surjective. Then,

$g\left(x\right)=1+x-\left[x\right]$ and $f\left(x\right)=\left\{\begin{array}{c}-3; x<0\\ 0; x=0\\ 5; x>0\end{array}\right.$, then $f\left(g\right(x\left)\right)$ is (where $\left[·\right]$ represents greatest integer function)

Let $f:R\to R$ be defined by $f\left(x\right)=2x^2-5$ and $g:R\to R$ by $g\left(x\right)=\frac{x}{x^2+1}$, then $gof$ is _____ .

For $x\in \left(0, \frac{3}{2}\right)$, let $f\left(x\right)=\sqrt{x}$, $g\left(x\right)=\tan x$ and $h\left(x\right)=\frac{1-x^2}{1+x^2}$. If $\varphi \left(x\right)=\left(\left(hof\right)og\right)\left(x\right)$ then $\varphi \left(\frac{\mathrm{\pi }}{3}\right)=$

If $f\left(x\right)=x^2-1$ and $g\left(x\right)=\left|2x+3\right|$, then $fog\left(3\right)-gof\left(-3\right)=$

If $f\left(x\right)=7x-6, g\left(x\right)=3x^2$ and $h\left(x\right)=8x$, then prove that $\left(f\circ g\right)\circ h\left(-4\right)=f\circ \left(g\circ h\right)\left(-4\right)$.

If $f\left(x\right)=3x+2$ and $g\left(x\right)=4x-5$, then $f\circ g\left(x\right)=g\circ f\left(x\right)$.

$f\left(x\right)=3x-12$ and $g\left(x\right)=\sqrt{x}$. Find the domain and range of $f\circ g\left(x\right)$.