Prove that the function f: R → R : f(x)= 2x is one-one and onto.

1.	Prove that the function f: R → R : f(x)= 2x is one-one and onto.
Answer» To prove: function is one-one and onto Given: f: R → R : f(x)= 2x We have, f(x) = 2x For, f(x₁) = f(x₂) ⇒ 2x₁ = 2x₂ ⇒ x₁ = x₂ When, f(x₁) = f(x₂) then x₁= x₂ ∴ f(x) is one-one f(x) = 2x Let f(x) = y such that \(y\in R\) ⇒ y = 2x \(\Rightarrow x=\frac{y}{2}\) Since \(y\in R\) \(\Rightarrow \frac{y}{2}\in R\) ⇒ x will also be a real number, which means that every value of y is associated with some x ∴ f(x) is onto Hence Proved

Answer»

To prove: function is one-one and onto

Given: f: R → R : f(x)= 2x

We have,

f(x) = 2x

For, f(x₁) = f(x₂)

⇒ 2x₁ = 2x₂

⇒ x₁ = x₂

When, f(x₁) = f(x₂) then x₁= x₂

∴ f(x) is one-one

f(x) = 2x

Let f(x) = y such that \(y\in R\)

⇒ y = 2x

\(\Rightarrow x=\frac{y}{2}\)

Since \(y\in R\)

\(\Rightarrow \frac{y}{2}\in R\)

⇒ x will also be a real number, which means that every value of y is associated with some x

∴ f(x) is onto

Hence Proved

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