Question

What is the domain and range of `f(x) = -7(x - 2)^2 - 9`?

Answer 1

See below.

Explanation:

`-7(x-2)^2-9`

This is a polynomial, so its domain is all `RR`.

This can be expressed in set notation as:

`{x in RR}`

To find the range:

We notice that the function is in the form:

`color(red)(y=a(x-h)^2+k`

Where:

`bbacolor(white)(88)`is the coefficient of `x^2`.

`bbhcolor(white)(88)` is the axis of symmetry.

`bbkcolor(white)(88)` is the maximum or minimum value of the function.

Because `bba` is negative we have a parabola of the form, `nnn`.

This means `bbk` is a maximum value.

`k=-9`

Next we see what happens as `x-> +-oo`

as `x->oo` , `color(white)(8888)-7(x-2)^2-9->-oo`

as `x->-oo` , `color(white)(8888)-7(x-2)^2-9->-oo`

So we can see that the range is:

`{y in RR | -oo < y <= -9}`

The graph confirms this:

graph{-7x^2+28x-37 [-1, 3, -16.88, -1]}