Solution to Question 1

The composition (f _{o} g) (x) of
f(x) = x + 1 , g(x) = 3x

is given by
(f _{o} g) (x) = f(g(x)) = g(x) + 1 = 3x + 1

Since the domain of both functions is the set of all real numbers, the composition (f _{o} g) (x) also have the set of all real numbers as its domain.
Solution to Question 2

The composition (f _{o} g) (x) of
f(x) = x^{ 2} + 1 , g(x) = √(2 x)

is given by
(f _{o} g) (x) = f(g(x))
= g(x)^{ 2} + 1
= [ √(2 x) ]^{ 2} + 1
= 2 x + 1

To find the domain of the composition of the two functions, we proceed as follows: x must be in the domain of g.
2 x ≥ 0 which is equivalent to x ≥ 0

g(x) must be in the domain of f(x). The domain of f is the set of all real numbers. The condition x ≥ 0 makes g(x) real and is therefore in the domain of f. Hence the domain of the composition is the set of all values defined by the interval
[0 , + ∞)
Solution to Question 3

The composition (f _{o} g) (x) of
f(x) = √( x + 1) , g(x) = x^{ 2}  8

is given by
(f _{o} g) (x) = f(g(x))
= √( g(x) + 1)
= √( (x^{ 2}  8) + 1)
= [ √(9  x^{ 2}) ]

The domain of g is the set of real numbers. Hence to find the domain of the composition, x must satisfy the condition
9  x^{ 2} ≥ 0

The solution set of the above inequality is also the domain given by the interval
[3 , 3]
Solution to Question 4

Given
f(x) = 2x + 1 and g(x) = x^{ 2} ,

(f _{o} g) (2) is calculated as follows
(f _{o} g) (2) = f(g(2))
= f(4) = 9
Solution to Question 5

Given
f(x) = 1 / (x + 1) and g(x) = 1 / (x  1) ,

(f _{o} g) (0) is calculated as follows
(f _{o} g) (0) = f(g(0))
= f(1) = undefined since x = 1 makes the denominator of f equal to zero.
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