Show that the function $f:R\to R:f\left(x\right)=1+x^2$ is many-one into.

Let $f:R\to R:f\left(x\right)=\frac{2x-7}{4}$ be an invertible function. Find $f^{-1}$

Let $f:R\to R:f\left(x\right)=10x+3$. Find $f^{-1}$.

$f:R\to R:f\left(x\right)=\left\{\begin{array}{c}1, \mathrm{if} x \mathrm{is} \mathrm{rational}\\ -1 \mathrm{if} x \mathrm{is} \mathrm{rational}\end{array}\right.$.  Show that $f$ is many-one and into.

Let $f\left(x\right)=x+7$ and $g\left(x\right) = x -7, x \in R$. Find $(f \circ g) \left(7\right).$

Let $f : R \to R$and $g : R \to R$ defined by $f\left(x\right)=x^2$ and $g\left(x\right) = (x + 1).$ Show that $g \circ f \ne f \circ g.$

Let $f:R\to R:f\left(x\right)={\left(3-x^3\right)}^{1/3}$. Find $f\circ f.$

Let $f : R \to R : f\left(x\right) = 3x + 2$, find $f\left(f\left(x\right)\right)$.

Let $f=\left\{\right(1,2),(3,5),(4,1)\}$ and $g=\{(1,3),(2,3),(5,1))$. Write down $g o f$.