Let f : R → R be the function defined by f(x) = 2x – 3 ∀ x ∈ R. w… - Vidyadip

Question

Let f : R → R be the function defined by f(x) = 2x – 3 ∀ x ∈ R. write f^-1.

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Answer

Given f(x) = 2x - 3 ∀ x ∈ R

Now, Let a, b ∈ R such that

f(a) = f(b)

⇒ 2a - 3 = 2b - 3

⇒ a = b

⇒ f(x) is One-One.

Also, If x, y ∈ R such that

f(x) = y ⇒ 2x - 3 = y

$\Rightarrow\ \text{x}=\frac{\text{y}+3}{2}=\text{g}(\text{y})\ \forall\ \text{y}\in\text{R}$

⇒ f(x) is Onto and therefore is bijective implies f(x) has an inverse

Let f^-1 denote the inverse of f(x) then,

$\text{f}^{-1}\text{x}=\text{g}(\text{x})=\frac{\text{x}+3}{2}\ \forall\ \text{x}\in\text{R}$

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