To find the inverse function [tex]\( b(a) \)[/tex] that takes the area of the trapezoid as input and returns the length of the other base, we start with the given function:
[tex]\[ a(b) = 12 \cdot \frac{b + 9}{2} \][/tex]
First, we simplify this equation:
[tex]\[ a(b) = 6 \cdot (b + 9) \][/tex]
[tex]\[ a(b) = 6b + 54 \][/tex]
To find the inverse function, we need to solve this equation for [tex]\( b \)[/tex] in terms of [tex]\( a \)[/tex].
First, set [tex]\( a \)[/tex] as the output of the function:
[tex]\[ a = 6b + 54 \][/tex]
Next, solve for [tex]\( b \)[/tex]:
1. Subtract 54 from both sides:
[tex]\[ a - 54 = 6b \][/tex]
2. Divide both sides by 6:
[tex]\[ b = \frac{a - 54}{6} \][/tex]
Simplifying further, we get:
[tex]\[ b = \frac{a}{6} - 9 \][/tex]
So, the inverse function [tex]\( b(a) \)[/tex] is:
[tex]\[ b(a) = \frac{a}{6} - 9 \][/tex]
Therefore, the correct choice is:
C. [tex]\( b(a) = \frac{a}{6} - 9 \)[/tex]
