Question

If \(f\) is a relation from the set of positive real numbers to the set of positive real numbers defined by \[ f(x)=3x^2-2, \] then \(f\) is

• A. one-one but not onto
• B. onto but not one-one
• C. a bijection
• D. not a function

Hint:

When checking whether a rule defines a function from a set \(A\) to a set \(B\), always verify that the image of every element of \(A\) lies inside \(B\). If even one image lies outside the codomain, the given rule is not a function from \(A\) to \(B\).

Answer 1

The Correct Option is D

Step 1: Check whether the output always stays in the codomain.
The codomain is the set of positive real numbers. For \(x = 0.5\), \[f(0.5) = 3(0.25) - 2 = 0.75 - 2 = -1.25,\] which is not a positive real number.

Step 2: Conclude.
Since there exist positive real inputs whose image under \(f\) lies outside the set of positive reals, \(f\) does not map into its stated codomain, so it is not a function from positive reals to positive reals.
\[\boxed{f \text{ is not a function}}\]