Question 1
Find (f o g)(2) given that
f(x) = 3x + 2 and g(x) = x  4
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
Find (f o g)(x) and the domain of f o g given that
f(x) = (x  1) / (x + 2) and g(x) = (x + 1) / (x  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
Find the composition (f o g)(x) and the domain of f o g given that
f(x) = x^{ 2} + 2 and g(x) = √(x  2)
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.
Find the composition (f o g)(x) and its domain given f and g below:
a) f(x) = 2x^{ 3} + x  1 and g(x) = x^{ 2}
b) f(x) =  x^{ 2}  4  and g(x) = x  1
c) f(x) = x^{ 2}  5 and g(x) = √(x + 5)
d) f(x) = ln x and g(x) = √(x + 5)
e) f(x) = sin x and g(x) = x  2
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: ( ∞ , + ∞)
More References and linksComposition of Functions
A video with more on composite functions is available.
Composition of Functions Questions a
Questions on Composite Functions with Solutions.
