Question

Let A = {x ∈ R : x is not a positive integer} Define a function f :A → R as f(x) = 2x/(x - 1)
then f is
(A) injective but nor surjective (B) not injective
(C) surjective but not injective (D) neither injective nor surjective

Answer 1

Define a function f :A → R as f(x) = 2x/(x - 1)  then f is injective but not surjective

We know that a One - One or Injective function is a function in which every element of range of function corresponds to exactly one elements.

• In the graph of a function, if its strictly increasing or decreasing for all values of x then f is one - one

• In other words, if we consider the graph of the function, a constant y value or line parallel to x - axis cut the curve at most one point.

Given,

• $f(x) = \frac{2x}{x-1}$

• $\\f(x) =2 \left (1 + \frac{1}{x-1} \right )$
• $\\ f'(x) = -\frac{2}{(x-1)^2}$

From the derivative function its clear that the function  is strictly decreasing.

Therefore each x has different y values.

Therefore f is one-one i.e injective but not surjective.

Answer 2

The function f defined as f :A → R as f(x) = 2x/(x - 1) is

(A) injective but nor surjective

• We know that the denominator for a fraction should never be zero.
• Hence in the given function f, x is not equal to 1.
• Let us assume 2x/(x - 1) = y.
• 2x = xy-y
• y = x(y-2)
• x = y/y-2
• From this equation we know that y is not equal to 2.
• However, as f :A → R which inclues 2 in the mapped set, the function is injective but nor surjective.