Let a function $$f : (0, \infty) \to (0, \infty)$$ be defined by $$f(x) = \left|1 - \frac{1}{x}\right|$$. Then f is:

First remember the definition of a function: every element of the domain must be assigned an image that actually lies inside the codomain. The codomain fixed by the question is $$(0,\infty),$$ i.e. the set of strictly positive real numbers.

Now we examine the value of the given rule at the point $$x=1.$$ Substituting in the formula

$$f(x)=\left|1-\frac1x\right|,$$

we obtain

$$f(1)=\left|1-\frac11\right|=\left|1-1\right|=\left|0\right|=0.$$

But $$0\notin(0,\infty).$$ Therefore the image of the point $$x=1$$ does not belong to the declared codomain. This violates the very definition of a function from $$(0,\infty)$$ to $$(0,\infty).$$

Because the mapping is not even well-defined with the announced codomain, questions about injectivity or surjectivity are meaningless: the object under discussion fails to be a bona-fide function in the first place. Consequently none of the descriptions “injective only”, “not injective but surjective”, or “neither injective nor surjective” can be accepted.

Hence, the correct answer is Option D.