Question:
Let \( * \) be defined on \( \mathbb{R} – \{-1\} \) by:
\[ a * b = a + b + ab \]
Find the inverse of \( a \).
Options:
- (a) \( -a \)
- (b) \( -\frac{a}{a+1} \)
- (c) \( \frac{1}{a} \)
- (d) \( a^2 \)
Solution:
Step 1: Find identity element
Let identity be \( e \), then:
\[ a * e = a \Rightarrow a + e + ae = a \]
\[ e + ae = 0 \Rightarrow e(1 + a) = 0 \]
Since \( a \neq -1 \), we get:
\[ e = 0 \]
—Step 2: Find inverse of \( a \)
Let inverse be \( x \), then:
\[ a * x = 0 \]
\[ a + x + ax = 0 \]
\[ x(1 + a) = -a \Rightarrow x = -\frac{a}{a+1} \]
—Final Answer:
\[ \boxed{-\frac{a}{a+1}} \]
Correct Option: (b)