How do you find (f o g)(x) and its domain, (g o f)(x) and its domain, (f o g)(-2) and (g o f)(-2) of the following problem #f(x) = 2x + 3#, #g(x) = 3x -1#?

Question

How do you find (f o g)(x) and its domain, (g o f)(x) and its domain, (f o g)(-2) and  (g o f)(-2) of the following problem #f(x) = 2x + 3#, #g(x) = 3x -1#?

Caleb Alexander · Accepted Answer

undefined To find (f o g)(x) and its domain:

1. Substitute g(x) into f(x):  
   (f o g)(x) = f(g(x)) = f(3x - 1) = 2(3x - 1) + 3 = 6x - 2 + 3 = 6x + 1.

2. The domain of (f o g)(x) is the set of all real numbers since there are no restrictions on the domain of the composed function.

To find (g o f)(x) and its domain:

1. Substitute f(x) into g(x):  
   (g o f)(x) = g(f(x)) = g(2x + 3) = 3(2x + 3) - 1 = 6x + 9 - 1 = 6x + 8.

2. The domain of (g o f)(x) is the set of all real numbers since there are no restrictions on the domain of the composed function.

To find (f o g)(-2) and (g o f)(-2):

1. Substitute x = -2 into the composed functions:
   (f o g)(-2) = f(g(-2)) = f(3(-2) - 1) = f(-7) = 2(-7) + 3 = -14 + 3 = -11.
   (g o f)(-2) = g(f(-2)) = g(2(-2) + 3) = g(-1) = 3(-1) - 1 = -3 - 1 = -4.

Sophie Adams · Answer

See answer below

This is an assembly of various functions. #f(x)=2x+3#, #=>#, #D_f(x)=RR# #g(x)=3x-1#, #=>#, #D_g(x)=RR# #(fog)(x)=f(g(x))=f(3x-1)=2(3x-1)+3# #=6x-2+3=6x+1# The domain is #D_(fog)(x)=RR# #(fog)(-2)=6*-2+1=-11# #(gof(x))=g(f(x))=g(2x+3)=3(2x+3)-1=6x+9-1=6x+8# The domain is #D_(gof(x))=RR# #(gof(-2))=6*-2+8=-4#