Question

Given the function f(x)=3x -2 and g(x)= x+2/3 complete parts A and B.

A. Find f(g(x)) and g(f(x)). Include your work in your final answer.
B. Use complete sentence's to explain the relationship that exists between the composition of the functions. F(g(x)) and g(f(x)).

Answer 1

A

f(g(x)) = f(
`(x+2)/(3)`) = 3(
`(x+2)/(3)`) - 2 = x + 2 - 2 = x

g(f(x)) = g(3x - 2) =
`(3x-2+2)/(3)` =
`(3x)/(3)` = x

B

Since both composite functions f(g(x)) and g(f(x)) equal x

This indicates that the functions f(x) and g(x) are inverse functions

Answer 2

First, I should point out that g(x) should be written as g(x)=(x+2)/3, otherwise the problem is confusing.

`f(x)=3x-2 \enspace g(x)=(x+2)/(3)`

(A)
`f(g(x))=3((x+2)/(3))-2=x\\g(f(x))=(3x-2+2)/(3)=x`

(B) Since
`f(g(x))=x` and
`g(f(x))=x`, it holds that

`f(g(x))=g(f(x))` for all x. This means the composed functions are *identical*