Let f(x)=e^{x}-e^{-x}=2sinh(x), where sinh(x) is the hyperbolic sine function .
Then f'(x)=e^{x}+e^{-x}=2cosh(x), where cosh(x) is the hyperbolic cosine function.
Clearly f'(x)>0 for all x\in RR, implying that f(x) is strictly increasing over RR, and also over [-1,2].
Therefore, the minimum value of f over [-1,2] is f(-1)=2sinh(-1)=e^{-1}-e^{1}=(1-e^{2})/e approx -2.3504 and the maximum value of f over [-1,2] is f(2)=2sinh(2)=e^{2}-e^{-2}=(e^{4}-1)/(e^{2}) approx 7.2537.
This implies that the range is the closed interval =[2sinh(-1),2sinh(2)]=[(1-e^{2})/e,(e^{4}-1)/(e^{2})]approx [-2.3504,7.2537].