ANSWER
Domain: All real numbers
Range:
[tex][2, \infty )[/tex]
EXPLANATION
The given function is
[tex]y = 3 {x}^{2} - 6x + 5[/tex]
To find the domain and range of the given function, we complete the square.
[tex]y = 3 ({x}^{2} - 2x )+ 5[/tex]
[tex]y = 3 ({x}^{2} - 2x + 1) + 3( - 1)+ 5[/tex]
[tex]y = 3 ({x - 1)}^{2} - 3+ 5[/tex]
[tex]y = 3 ({x - 1)}^{2} + 2[/tex]
The vertex is at (1,2).
The given function is a polynomial and all polynomial functions are defined everywhere.
The domain is all real numbers.
The parabola opens upwards and have vertex at (1,2). Hence the minimum y-value is 2.
The range is
[tex][2, \infty )[/tex]
