A binary operation \circ on the set of real numbers \mathbb{R} is defined by a \circ b = a… — JAMB UTME

A binary operation $\circ$ on the set of real numbers $\mathbb{R}$ is defined by $a \circ b = a + b + 2ab$ . If the identity element of the operation is $e$ , what is the value of $x$ such that the inverse of $x$ under $\circ$ is equal to $2$ ?

Answer: -0.4

Answer

The value of

x

-0.4

First find the identity element

e

by solving

a \circ e = a

, which yields

a + e + 2ae = a \implies e(1 + 2a) = 0 \implies e = 0

. Next, by definition of an inverse,

x \circ x^{-1} = e

. Substituting

x^{-1} = 2

and

e = 0

gives

x \circ 2 = 0

. Expanding this using the binary operation rule yields

x + 2 + 2(x)(2) = 0 \implies 5x + 2 = 0 \implies x = -0.4

Step-by-Step Solution

Find the identity element

e

of the operation

\circ

e = 0

By definition of identity element,

a \circ e = a \implies a + e + 2ae = a

, which simplifies to

e(1 + 2a) = 0

, giving

e = 0

Set up the inverse equation using

x^{-1} = 2

x \circ 2 = 0

The inverse of

x

satisfies

x \circ x^{-1} = e

. Since

x^{-1} = 2

and

e = 0

x \circ 2 = 0

Solve for

x

x = -0.4

Expanding

x \circ 2 = 0

gives

x + 2 + 4x = 0 \implies 5x = -2 \implies x = -0.4

Key Concept

Identity and Inverse Elements in Binary Operations

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