Question 1

Solution to question 1
note that
(f o g)(-2) = f( g(-2) )
evaluate g(-2).
g(-2) = |-2 - 4| = 6
evaluate f( g(-2) ).
f( g(-2) ) = f(6) = -3*6 + 2 = -16
conclusion:
(f o g)(-2) = -16

Question 2

Solution to question 2
First find (fog)(x)
(f o g)(x) = f( g(x) ) = (g(x) - 1)/(g(x) + 2)
=[ (x + 1)/(x - 2) - 1 ] / [ (x + 1)/(x - 2) + 2 ]
= 3 / (3x - 3)

First find domain of f and g
domain of f : x not equal to -2
domain of g : x not equal to 2
g(x) has to be in the domain of f.
g(x) not equal to -2
solve for x the equation g(x) = -2
(x + 1)/(x - 2) = -2
x + 1 = -2x + 4
3x = 3
x = 1
for g(x) to be different from - 2, x has to be different from 1.
conclusion: The domain of f o g is: (- ∞ , 1) U (1 , 2) U (2 , + ∞)

Question 3

Solution to question 3
First find (f o g)(x)
(f o g)(x) = f( g(x) ) = g(x) ² + 2
= ( √(x - 2) ) ² - 2
= x

First find domain of f and g
domain of f : all real numbers
domain of g : x - 2 ≥ 0 ; x ≥ 2
Since the domain of f is all real numbers, we have to make sure that x is in the domain of g so that g has a real value.
conclusion: The domain of f o g is: [2 , +∞)

More Questions on Composition of Functions.

Answers to Above Questions.
Find the composition (f o g)(x) and its domain given f and g below:
a) (f o g)(x) = 2 x ⁶ + x ² -1    ,    domain: (-∞ , + ∞)
b) (f o g)(x) = | x ² - 2x - 3 |    ,    domain: (-∞ , + ∞)
c) (f o g)(x) = x    ,    domain: [ -5 , + ∞)
d) (f o g)(x) = (1 / 2) ln (x + 5)    ,    domain: [ -5 , + ∞)
e) (f o g)(x) = sin (x - 2)    ,    domain: (- ∞ , + ∞)

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