Which one of the following functions is one-one? 

We observe that $f\left(1\right)=3$and $f(-1)=3\text{. }$

$\therefore 1\ne -1$but $f\left(1\right)=f(-1)$

So, f is not a one-one function. 

Clearly, $1,-1\in Z$ such that $g\left(1\right)=1$and $g(-1)=(-1{)}^4=1$

i.e. $1\ne -1\text{ but }g\left(1\right)=g(-1)$

So, g is not a one-one function.  

Let x, $y\in \mathrm{ℝ}$ be such that

$h\left(x\right)=h\left(y\right)\Rightarrow x^3+4=y^3+4\Rightarrow x^3=y^3\Rightarrow x=y$

$\therefore h:R\to R$  is a one-one function. 

We observe that $w,w^2\in C$ (w is a cube root of unity) such that $w\ne w^2$but $\varphi \left(w\right)=w^3+4=5$ and  $\varphi \left(w^2\right)=w^6+4=5\text{. }$

i.e.  $w\ne w^2\text{ but }\varphi \left(w\right)=\varphi \left(w^2\right)$

So,$\varphi$ is not a one-one function. 

Graphical Method: Following steps may be used to check the

 Injectivity of a function from its graph: 

STEP I Draw the curve y = f (x).  

STEP II If every line parallel to x-axis cuts the curve exactly at one point i.e. the curve has no bend or turn, then f (x) is one-one. Otherwise it is a many-one function. If there exists a line parallel to x-axis cutting the curve y = f (x) at more than one point, then f (x) is a many one function.