How do you find the domain and range of #-x^2 + 4x -10#?

Question

2662 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-06-26T07:06:00

Domain: #(-oo, +oo)#
Range: #(-oo, -6]#

#f(x) -x^2+4x-10#

#f(x)# is defined #forall x in RR#

Hence the domain of #f(x)# is #(-oo, +oo)#

#f(x)# is a parabola of the form: #ax^2+bx+c#

Since the coefficient of #x^2<0#, #f(x)# will have a maximum value where #x=(-b)/(2a)#

#(-b)/(2a) =(-4)/(-2) = 2#

#:. f_"max" = f(2) = -2^2+4*2-10 = -4+8-10 = -6##

Since #f(x)# has no finite lower bound the range of #f(x)# is #(-oo, -6]#

We can see these from the graph of #f(x)# below.
graph{-x^2+4x-10 [-44.87, 37.33, -28.45, 12.64]}