Find the inverse of p under the binary operation * defined by p*q = p + q - pq, where p and q are real numbers and zero is the identity
A. p
B. p -1
C. p/(p-1)
D. p/(p+1)
To find the inverse of p under the binary operation *, we need to find a number x such that p * x = x * p = 0, where 0 is the identity element.
Let's assume the inverse of p is x. Then we have: p * x = p + x - px = 0.
We can rearrange this equation to solve for x:
p + x - px = 0
x - px = -p
x(1 - p) = -p
x = -p / (1 - p)
Therefore, the inverse of p is x = -p / (1 - p).
The correct answer is D. p / (p + 1).
To find the inverse of p under the binary operation *, we need to find another real number q such that p * q = q * p = 0.
Given that p * q = p + q - pq, we can set this equation equal to 0:
p + q - pq = 0
Rearranging the terms, we get:
q - pq = -p
Factoring out q, we have:
q(1 - p) = -p
Dividing both sides of the equation by (1 - p), we obtain:
q = -p / (1 - p)
Therefore, the inverse of p under the binary operation * is:
q = -p / (1 - p)
Hence, the correct answer is option D: p / (p + 1).