Let f: N → R be the function defined by f(x) = (2x−1)/2 and g: Q → R be another function defined by g (x) = x + 2. Then (g 0 f) 3/2 is

The correct option is "None of these".

To understand why this is the correct choice, let us analyze the definition of the composite function and the domains of the individual functions given in the problem.

We are given two functions:

1. A function

$f:N\to R$

defined by

$f\left(x\right)=\frac{2x-1}{2}$

where the domain of the function is the set of natural numbers

$N=\{1,2,3,\dots \}$

and the codomain is the set of real numbers

$R$

2. Another function

$g:Q\to R$

defined by

$g\left(x\right)=x+2$

where the domain is the set of rational numbers

$Q$

We need to evaluate the composite function

$(g\circ f)\left(\frac{3}{2}\right)$

By definition, the composition is evaluated as:

For the composition to be defined at a value, the input value must belong to the domain of the inner function. Here, the inner function is

$f$

which has the domain

$N$

Our input value is:

$x=\frac{3}{2}=1.5$

Since

$1.5$

is a fractional value and not a positive integer, it does not belong to the set of natural numbers:

$\frac{3}{2}\notin N$

Because the input does not lie in the domain of the function

$f$

the expression

is undefined. Consequently, the composition

cannot be evaluated and is undefined. Therefore, the correct choice is "None of these".