To find the inverse function [tex]\( b(a) \)[/tex], we need to express [tex]\( b \)[/tex] in terms of [tex]\( a \)[/tex].
The function given is:
[tex]\[ a(b) = 12 \cdot \frac{b + 9}{2} \][/tex]
First, let's rewrite it for clarity:
[tex]\[ a = 12 \cdot \frac{b + 9}{2} \][/tex]
The goal is to solve this equation for [tex]\( b \)[/tex] in terms of [tex]\( a \)[/tex].
1. Start by isolating the fraction on the right-hand side:
[tex]\[ a = 12 \cdot \frac{b + 9}{2} \][/tex]
2. Multiply both sides by 2 to get rid of the denominator:
[tex]\[ 2a = 12(b + 9) \][/tex]
3. Next, divide both sides by 12 to isolate [tex]\( b + 9 \)[/tex]:
[tex]\[ \frac{2a}{12} = b + 9 \][/tex]
4. Simplify the left-hand side:
[tex]\[ \frac{a}{6} = b + 9 \][/tex]
5. Finally, subtract 9 from both sides to solve for [tex]\( b \)[/tex]:
[tex]\[ b = \frac{a}{6} - 9 \][/tex]
Therefore, the equation representing the inverse function [tex]\( b(a) \)[/tex], which takes the trapezoid's area as input and returns the length of the other base, is:
[tex]\[ \boxed{b(a) = \frac{a}{6} - 9} \][/tex]
Thus, the correct choice is A.
