Question

What is the range of `f(x)=–5-2(x+3)^2`?

Answer 1

The range is `{y|y<="-5"}`.

Explanation:

The range of a function is simply all the possible outputs that function can give.

Mathematically speaking, a number `y` is in the range of a function `f` when we can find an `x` such that `f(x)=y`.

There are a couple of ways to find the range of a function. For `f(x)="-5"-2(x+3)^2`, the easiest way is to see that `f` is a certain type of function—a parabolic function. As such, there is no limitation on the `x`-values we can input (i.e. the domain is `RR`), but there is an easy-to-see limitation on the output (or `y`) values `f(x)` can give.

Take a look at that squared bit—the `(x+3)^2`. We know this can't be less than 0, because the square of a number is always positive (or 0, if we're squaring 0).

So `(x+3)^2` is no less than 0. Meaning, `2(x+3)^2` is also no less than 0. Then `-2(x+3)^2` is no more than 0, and thus `"-"5-2(x+3)^2` is no more than -5.

In math:

`color(white)(f(x)="-5"-2)(x+3)^2>=0`
`color(white)(f(x)="-5"-)2(x+3)^2>=0`
`color(white)(f(x)="-5")-2(x+3)^2<=0" "`(note the change to `<=`)
`f(x)="-5"-2(x+3)^2<="-5"`

The end result: `f(x)<="-5"`.
And so our range is "all numbers `y` such that `y` is at most -5", or in math:

`{y|y<="-5"}`.

Bonus:

The shortcut to finding the range of a parabolic equation `f(x)=a(x-h)^2+k` is to use this:

Range of `f(x)` is `{({y|y>=k}" when "a>0),({y|y<=k}" when "a<0):}`

Just choose the correct option depending on the value of `a`. (In this example, `a` was less than 0, so we would choose `{y|y<=k}`, and plug in our `k`-value of -5.)