Question

What is the range of `y = 3x^2 - 7`?

Answer 1

Since your function is a quadratic and the quadratic coefficient is positively-signed, it has a single minimum value, and extends upwards towards `oo`.

Therefore, when you've found the smallest possible y value, that's your lower bound.

Since the second degree term is not horizontally shifted (that is, `(x+c)^2`, where `c=0`), the minimum is when `x=0`.

So, your range is:

`color(blue)([-7", " oo ))`

Answer 2

Another way is to see that

`f(x)+7=3x^2>=0`

`f(x)+7>=0`

`f(x)>=-7`

Hence the range is `R_f=[-7,+oo)`