Question #99878

1 Answer
Oct 10, 2017

Answer:

The domain of #f(x)# is #x in RR# and the range is #f(x) in (-oo,12.5]#
The domain of #f(x)# is #x in RR# and the range is #g(x) in [0,+oo)#

Explanation:

First part #f(x)#

#f(x)=12-2x-2x^2#

This is a polynomial function and it is defined over #RR#

Therefore,

The domain of #f(x)# is #x in RR#

Let #y=12-2x-2x^2#

Rewriting the equation

#2x^2+2x-12+y=0#

For this quadratic equation to have solutions, the discriminant #Delta>=0#

#Delta=b^2-4ac=(2)^2-4*(2)*(y-12)#

#4-8y+96>=0#

#8y<=100#

#y<=100/8#

The range is #y in (-oo,12.5]#

graph{12-2x-2x^2 [-26.34, 31.37, -7.27, 21.6]}

Second part *#g(x)#*

#g(x)=|f(x)|#

The domain of #g(x)# remains the same, #x in RR#

The range changes as all the negative values become positive

So,

the range is #g(x) in [0,+oo)#

graph{|12-2x-2x^2| [-29.1, 28.6, -1.96, 26.9]}