What is the domain and range of #F(x) = -2(x + 3)² - 5#?

1 Answer
Nov 22, 2015

Answer:

Domain: #(-oo,+oo) in RR#
Range: #(-oo,-5] in RR#

Explanation:

#F(x) = -2(x+3)^2-5# can be evaluated for all values of #x in RR#
so the Domain of #F(x)# is all #RR#

#-2(x+3)^2-5#
is a quadratic in vertex form with vertex at #(-3,-5)#
and the negative coefficient of #(x+3)^2# tells us that the quadratic opens downward;
therefore #(-5)# is a maximum value for #F(x)#

Alternative way of seeing this:
#(x+3)^2# has a minimum value of #0# (this is true for any squared Real value)
therefore
#-2(x+3)^2# has a maximum value of #0#
and
#-2(x+3)^2-5# has a maximum value of #(-5)#

Second alternative
consider the graph of this function:
graph{-2*(x+3)^2-5 [-17.42, 5.08, -9.78, 1.47]}