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

1 Answer
Alan P.
Oct 1, 2015

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)

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