How do you find the domain and range of #f(x)= -2x^2+8x-5#?

1 Answer
Dec 3, 2017

Answer:

Domain, # x in RR#
Range, # f(x) <= 3#

Explanation:

First we can conisder the domain, this is fairly simple, we must consider what values of #x # yields a valid value of #f(x)#, and we see for all values of #x#, #f(x)# is defined, and we can see that by a sketch; graph{-2x^2+8x-5 [-8.58, 11.42, -4.36, 5.64]}

To consider the range, we must cosnider all the values #f(x)# can take on, and by the sketch, we see the the max value of #f(x)# is 3, this is the vertex point, where the vertex point is defined as being;
#((-b)/(2a),f( (-b)/(2a) ) )# as we can prove this rather simply using stationary points and differential calculus, and we see the vertex point is #(2,3)#

So from there #f(x)# can take on any value lower than 3,
#=># #f(x) <= 3#