Example 2 Prove that the function f : R →
→
→
→
→ R
defined by f (x) = 2x + 5 is one-one Solution Note that a function f is one-one if
f (x1) = f (x2) ⇒ x1 = x2  (definition of one-one function)
Now,  given that
f (x1) = f (x2), i e , 2x1+ 5 = 2x2 + 5
⇒
2x1+ 5 – 5 = 2x2 + 5 – 5  (adding the same quantity on both sides)
Rationalised 2023-24
PROOFS IN MATHEMATICS
189
⇒
2x1+ 0 = 2x2 + 0
⇒
2x1 = 2x2 (using additive identity of real number)
⇒
1
2
2 x  =
2
2
2 x  (dividing by the same non zero quantity)
⇒
x1 = x2
Hence, the given function is one-one