Question:
Let \( * \) be defined on \( \mathbb{Q}^+ \) by:
\[ a * b = \frac{ab}{2} \]
Find the inverse of \( 8 \).
Options:
- (a) \( \frac{1}{8} \)
- (b) \( \frac{1}{2} \)
- (c) 2
- (d) 4
Solution:
Step 1: Find identity element
Let identity be \( e \), then:
\[ a * e = a \Rightarrow \frac{ae}{2} = a \]
\[ ae = 2a \Rightarrow e = 2 \]
Step 2: Find inverse of 8
Let inverse be \( x \), then:
\[ 8 * x = 2 \]
\[ \frac{8x}{2} = 2 \]
\[ 4x = 2 \Rightarrow x = \frac{1}{2} \]
—Final Answer:
\[ \boxed{\frac{1}{2}} \]
Correct Option: (b)