Question

What is the domain and range of `f(x) = x^2 + 4x – 6`?

Answer 1

Domain: `RR`
Range: `RR >= -10`

Explanation:

`f(x)=x^2+4x-6`
is valid for all Real values of `x`
and therefore the Domain is all Real values i.e. `RR`

To determine the Range, we need to find what values of `f(x)` can be generated by this function.
Probably the simplest way to do this is to generate the inverse relation. For this I will use `y` in place of `f(x)` (just because I find it easier to work with).

`y=x^2+4x-6`

Reversing the sides and completing the square:
`color(white)("XXX")(x^2+4x+4) - 10 = y`

Re-writing as a square and adding `10` to both sides:
`color(white)("XXX")(x+2)^2=y+10`

Taking the square root of both sides
`color(white)("XXX")x+2 = +-sqrt(y+10)`

Subtracting `2` from both sides
`color(white)("XXX")x= +-sqrt(y+10) -2`

Assuming that we are restricted to Real values (i.e. non-Complex), this expression is valid provided:
`color(white)("XXX")y>=-10`
`color(white)("XXXXXX")`(otherwise we would be dealing with the square root of a negative value)