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

A binary operation $\ast$ is defined on the set of real numbers $\mathbb{R}$ by $a \ast b = a + b - k$ , where $k$ is a constant. If the identity element of the operation is $4$ , what is the inverse of $7$ under this operation?

1Cevap
B
-1
C
-3
D
7

Cevap

The inverse of 7 under the binary operation is 1.

First, use the identity property a * e = a with e = 4 to determine that k = 4. Then, set 7 * x = 4 using the operational definition 7 + x - 4 = 4. Solving the linear equation x + 3 = 4 gives the correct inverse value of 1.

Adım Adım Çözüm

Determine the value of the constant k using the identity element property.

k = 4

By definition of identity element e, a * e = a. Given e = 4, substituting into the definition yields a + 4 - k = a, which simplifies to k = 4.

Write the full operational formula.

a * b = a + b - 4

Substitute k = 4 into the original rule a * b = a + b - k.

Solve for the inverse element of 7.

x = 1

Let x be the inverse of 7. By definition of inverse, 7 * x = e, so 7 + x - 4 = 4. Simplifying gives x + 3 = 4, hence x = 1.

Anahtar Kavram

Identity and inverse elements of a binary operation

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