Let # f(x)= (x+3)/(x^2 -4x-12)# and #g(x)=sqrt(3-x)#, how do you find each of the compositions and domain and range?

1 Answer
Sep 15, 2015

Answer:

As explained below.

Explanation:

#f@g(x) = ##(sqrt(3-x) +3)/((3-x)-4sqrt(3-x)-12)#

#g@f(x) = ##sqrt(3- (x+3)/(x^2-4x-12)#

f(x) can be written as #(x+3)/((x-6)(x+2))#,its domain would therefore be #(-oo,-2)U(-2,6)U(6,oo)#, or #{x in RR, x!= -2,6}#

For its range interchange x and y and solve the resultant quadratic equation for y. The result would be

y=#( (4x+1) +- sqrt((4x+1)(16x+1)))/(2x)#

This would mean #x<=-1/4# or #x>=-1/16#
Range would be# {y:RR, y<= -1/4 or y>= -1/16}#

For g(x), domain would be #{x:RR, x<=3}# so that all y values are real.

Range would be #{y: RR, y>=0}#