Question

What can be said about the domain of the function ( f ∘ g ) where ( f(y) = )4/y-2 ) and ( g(x) = )5/3x-1 )?

a) The domain is all real numbers except ( x = )1/3 ) and ( y = 2 ).

b) The domain is all real numbers except ( x = 1 ) and ( y = 2 ).

c) The domain is all real numbers except ( x = )(1)(3) ) and ( y = 0 ).

d) The domain is all real numbers except ( x = 1 ) and ( y = 0 ).

Answer 1

The domain of the composite function can be found by considering the domain restrictions of the individual functions. For (f ∘ g), the domain is all real numbers except x = 1/3 and y = 2.

Step-by-step explanation:

The domain of the composite function (f ∘ g) can be found by considering the domain restrictions of the individual functions f(y) and g(x).

f(y) = 4/(y-2) has a domain where the denominator (y-2) is not equal to zero. This means that y cannot be equal to 2.

g(x) = 5/(3x-1) has a domain where the denominator (3x-1) is not equal to zero. Solving for x, we find that x cannot be equal to 1/3.

The domain of (f ∘ g) will be the intersection of the domains of f and g. Thus, the domain is all real numbers except x = 1/3 and y = 2. Therefore, option a) The domain is all real numbers except (x = 1/3) and (y = 2) is correct.