, for this problem f(x)=3x^2-6x-7,what is the minimum value and where do it occurs, and what is this functions domain and range?

Answer 1

for any parabola, the min/max occurs where x = -b/2a

In your case, at x=6/6 = 1

domain for any polynomial is all reals

f(x) = 3x^2 - 6x - 7
= 3(x-1)^2 - 10

from this you can easily see that the min at x=1 is -10

so the range is all reals >= 10

Answer 2

By completing square,
f(x) = 3[x^2 -2x - 7/3]
= 3[x^2 - 2(x)(1) + (1)^2 - (1)^1 - 7/3]
= 3[(x - 1)^2 - 1 - 7/3]
= 3(x - 1)^2 - 3 - 7
= 3(x - 1)^2 - 10