If $f:[0,1]\to [-1,3]$ defined by $f\left(x\right)=x^2+x+1$ then $f\left(x\right)$ is 

The given function is $f\left(x\right)=x^2+x+1$

Given its domain is $\left[0,1\right]$ and its codomain is $\left[-1,3\right]$

The  graph of the funciton is as below

Observing the above graph, the function is one one

but it has no negative values in the range, so that codomain is not same as ragne

so that the funciton is not onto. 

Therefore, the funciton is only one to one.