An operation * is defined on the set of real numbers,R, by x *=x+y+3xy if * is commutative (a) find the Identity element of R under the operation (b) determine the inverse of x ER , under the operation (c) find the value of x ER for which the value exist
(a) To find the identity element of R under the operation *, we need to find a real number e such that for any x ∈ R, x * e = e * x = x.
Let's set x * e = x:
x * e = x + e + 3xe
Simplifying the equation, we get:
e + 3xe = 0
Factor out e:
e(1 + 3x) = 0
Since e cannot be 0 (because division by 0 is undefined), we set:
1 + 3x = 0
3x = -1
x = -1/3
Thus, the identity element e is -1/3.
(b) To find the inverse of x ∈ R under the operation *, we need to find a real number y such that x * y = y * x = e.
Let's set x * y = e:
x * y = -1/3
x + y + 3xy = -1/3
Rearranging the equation, we get:
3xy + x + y + 1/3 = 0
Using the quadratic formula, let's solve for y:
y = (-1 ± √(1 - 12x))/(6x)
Therefore, the inverse of x ∈ R under the operation * is y = (-1 ± √(1 - 12x))/(6x).
(c) To find the value of x ∈ R for which the value exists, we need to consider the square root term in the inverse equation.
1 - 12x ≥ 0
12x ≤ 1
x ≤ 1/12
Hence, for all x ≤ 1/12, the value exists in R.