A binary operation \star on the set of real numbers \mathbb{R} \setminus \{1\} is defined … — JAMB UTME

A binary operation $\star$ on the set of real numbers $\mathbb{R} \setminus \{1\}$ is defined by $a \star b = a + b - ab$ . If $y^{-1}$ denotes the inverse of an element $y$ under $\star$ , find the value of $x$ such that $(x \star 3)^{-1} = 2$ .

Answer: 0.5

Answer

The value of

x

0.5

To solve for

x

, first calculate the identity element

e = 0

using

a \star e = a

. Next, use

y \star y^{-1} = 0

to get the inverse formula

y^{-1} = \frac{y}{y - 1}

. Substituting

y = x \star 3 = 3 - 2x

into the inverse formula yields

\frac{3 - 2x}{2 - 2x} = 2

, which simplifies to

2x = 1

, giving

x = 0.5

Step-by-Step Solution

Find the identity element

e

of the operation

\star

e = 0

By definition of the identity element,

a \star e = a \implies a + e - ae = a \implies e(1 - a) = 0

, so

e = 0

for all

a \neq 1

Derive the general expression for the inverse element

y^{-1}

y

y^{-1} = \frac{y}{y - 1}

By definition of inverse element,

y \star y^{-1} = e \implies y + y^{-1} - y y^{-1} = 0 \implies y^{-1}(1 - y) = -y \implies y^{-1} = \frac{y}{y - 1}

Evaluate the inner operation

x \star 3

x \star 3 = 3 - 2x

Using the operational rule

a \star b = a + b - ab

, we obtain

x \star 3 = x + 3 - 3x = 3 - 2x

Set up and solve the equation

(x \star 3)^{-1} = 2

x = 0.5

Letting

y = 3 - 2x

, its inverse is

y^{-1} = \frac{3 - 2x}{(3 - 2x) - 1} = \frac{3 - 2x}{2 - 2x}

. Equating this to

2

gives

\frac{3 - 2x}{2 - 2x} = 2 \implies 3 - 2x = 4 - 4x \implies 2x = 1 \implies x = 0.5

Key Concept

Identity and Inverse Elements in Binary Operations

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