Question

Prove that the function f : R → R

defined by f(x) = 2x + 5 is one-one.​

Answer 1

Answer:

Solution :

Here, f(x)=2x

A function is a one-one function such that f(x1)=f(x2) only if x1=x2.

Here, f(x1)=2x1

f(x2)=2x2

If f(x1)=f(x2), then,

2x1=2x2⇒x1=x2

∴f(x) is one-one function.

Now, for any value of x, we have a different value of f(x). That means, f(x) is onto function also.

Explanation:

Answer 2

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Solution :

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)

⇒ 2x1+ 0 = 2x2 + 0

⇒ 2x1

= 2x2

(using additive identity of real number)

⇒ 1/2

2x = 22/2x (dividing by the same non zero quantity)

⇒ x1 = x 2

Hence, the given function is one-one