# Inverse Image and Inverse Function

## Inverse Image of an Element

Consider a function ƒ: A➞B. The inverse image of an element y∈B with respect to ƒ is defined as the set of elements in A which have y as their image. It is denoted by ƒ-1(y). $ƒ^{-1}(y)=\{x∈A: y=ƒ(x)\}$ Examples:

1. Let ƒ: A➞B be defined by the arrow diagram,

Then, the inverse of a under ƒ $ƒ^{-1}(a)=\{1,2\}$ and, ƒ-1(b) is a null set. $ƒ^{-1}(c)=3$

2. Let a function ƒ: R➞R be defined by ƒ(x)=x2. Then, $ƒ^{-1}(9)=\{-3,3\}$ ∵ 9 is the image of both -3 and 3. Also, $ƒ^{-1}(-1)=Φ$ ∵ There is no real number whose square is -1.

## Inverse Function

Let ƒ: A➞B be a bijective function. Since ƒ is onto, each element b∈B is an image of at least one element a∈A. But, ƒ is also one-one, so a is the only (or unique) element of A corresponding to the element b∈B. Inverse function, a function from B to A, associates each element b of B with a unique element a of A. In other words, a function of the type; $ƒ^{-1}:B\rightarrow A$ This function is known as the inverse function of ƒ.

Thus, if ƒ: A➞B is one-one and onto, then ther exists a function ƒ-1: B➞A called the inverse function of ƒ and is defined as a function in which every element of B associates with a unique element of A.

Examples:

1. Let ƒ: A➞B be one-one and onto function defined by the following diagram,

2. Let a function ƒ: R➞R be defined by $ƒ(x)=x^3$ This function is bijective. Hence, ƒ-1 exists and is defined by $ƒ^{-1}(x)=\sqrt{x}$ Q. Let a function ƒ: R➞R be defined by y=ƒ(x)=2x-3, x∈R. Find the inverse function.

Let x1, x2 ∈ R (domain). Then, $ƒ(x_1)=2x_1-3$ $ƒ(x_2)=2x_2-3$ Now, $ƒ(x_1)=ƒ(x_2)$ $\Rightarrow 2x_1-3=2x_2-3$ $\Rightarrow x_1=x_2$ ∴ ƒ is one-one.

Again, let k ∈ R. Then, $k=2x-3$ $\Rightarrow x=\frac{k+3}{2}∈R$ ∴ ƒ is onto.

Since, ƒ is one-one and onto, it is a bijective function and ƒ-1 exists. We have, $y=2x-3$ Interchanging the values of x and y, we get, $x=2y-3$ $y=\frac{x+3}{2}$ $\therefore ƒ^{-1}(x)=\frac{x+3}{2}$

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