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Difficulty: MediumBinary Operations

A binary operation ∗\ast is defined on the set of real numbers R\mathbb{R} by a∗b=a+b−4a \ast b = a + b - 4. If x−1x^{-1} represents the inverse of an element xx under the operation ∗\ast, what is the value of xx such that x∗3=10−1x \ast 3 = 10^{-1}?

  1. −1-1Answer
  2. B
    −5-5
  3. C
    −9-9
  4. D
    1110\frac{11}{10}

Answer

The value of xx is −1-1.
First, the identity element ee is determined using a∗e=aa \ast e = a, which gives a+e−4=a  ⟹  e=4a + e - 4 = a \implies e = 4. Next, the inverse 10−110^{-1} is calculated from 10∗10−1=410 \ast 10^{-1} = 4, yielding 10+10−1−4=4  ⟹  10−1=−210 + 10^{-1} - 4 = 4 \implies 10^{-1} = -2. Finally, setting x∗3=−2x \ast 3 = -2 gives x+3−4=−2  ⟹  x−1=−2  ⟹  x=−1x + 3 - 4 = -2 \implies x - 1 = -2 \implies x = -1.

Step-by-Step Solution

1
Find the identity element ee under the operation ∗\ast.
e=4e = 4
By definition of an identity element, a∗e=a  ⟹  a+e−4=a  ⟹  e=4a \ast e = a \implies a + e - 4 = a \implies e = 4.
2
Determine the inverse of 1010, denoted 10−110^{-1}, under ∗\ast.
10−1=−210^{-1} = -2
By definition of an inverse element, 10∗10−1=e  ⟹  10+10−1−4=4  ⟹  6+10−1=4  ⟹  10−1=−210 \ast 10^{-1} = e \implies 10 + 10^{-1} - 4 = 4 \implies 6 + 10^{-1} = 4 \implies 10^{-1} = -2.
3
Substitute 10−1=−210^{-1} = -2 into the given equation x∗3=10−1x \ast 3 = 10^{-1} and solve for xx.
x=−1x = -1
Using the definition of the operation, x∗3=x+3−4=x−1x \ast 3 = x + 3 - 4 = x - 1. Setting x−1=−2x - 1 = -2 yields x=−1x = -1.

Key Concept

Identity and Inverse Elements in Binary Operations
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Topics

Binary Operations

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