The equation is a parabola with the coefficient of term squared x negative, so the range is from - infinity to the y-value of the vertex.
We need to find the y-value of the vertex:
[tex]\begin{gathered} \text{The general form of a parabola is:} \\ y=ax^2+bx+c \\ T\text{he x-value of the vertex is:} \\ x_v=-\frac{b}{2a} \\ \text{In this case, a=-1, b=4 and c=-1, so}\colon \\ x_v=-\frac{4}{2\cdot(-1)}=2 \\ y_v=-2^2+4\cdot2-1 \\ y_v=-4+8-1=3 \end{gathered}[/tex]
So, the Range of the function is (-infinity, 3]
