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| 1. |
Prove that the function f : R → R defined by f (x) = 2x + 5 is one-one. |
| Answer» N A function is one-one if f(x1) = f(x2) ⇒ x1 = x2.Using this we have to show that “2x1+ 5 = 2x2 + 5” ⇒ “x1 = x2”. This is of the formp ⇒ q, where, p is 2x1+ 5 = 2x2+ 5 and q : x1 = x2. We have proved this in Example 2of “direct METHOD”.We can also prove the same by using contrapositive of the STATEMENT. Nowcontrapositive of this statement is ~ q ⇒ ~ p, i.e., contrapositive of “ if f (x1) = f (x2),then x1 = x2” is “if x1 ≠x2, then f(x1) ≠ f (x2)”.Now x1 ≠ x 2⇒ 2x1 ≠ 2x2⇒ 2x1+ 5 ≠ 2x2 + 5⇒ f (x1) ≠ f (x2).Thank you | |