To find the inverse function \( b(a) \) from the given area function \( a(b) \), we need to express \( b \) in terms of \( a \). Let's start with the given function:
[tex]\[
a(b) = 12 \cdot \frac{b + 9}{2}
\][/tex]
First, we simplify the equation:
[tex]\[
a(b) = 6 \cdot (b + 9)
\][/tex]
Next, we solve for \( b \) in terms of \( a \).
1. Start with the equation:
[tex]\[
a = 6 \cdot (b + 9)
\][/tex]
2. Divide both sides by 6:
[tex]\[
\frac{a}{6} = b + 9
\][/tex]
3. Subtract 9 from both sides:
[tex]\[
\frac{a}{6} - 9 = b
\][/tex]
Therefore, the inverse function \( b(a) \) is:
[tex]\[
b(a) = \frac{a}{6} - 9
\][/tex]
The correct answer is:
A. [tex]\( b(a) = \frac{a}{6} - 9 \)[/tex]
