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The function [tex]a(b)[/tex] relates the area of a trapezoid with a given height of 12 and one base length of 9 to the length of its other base.

It takes as input the other base value, and returns as output the area of the trapezoid.
[tex]\[ a(b) = 12 \cdot \frac{b+9}{2} \][/tex]

Which equation below represents the inverse function [tex]b(a)[/tex], which takes the trapezoid's area as input and returns as output the length of the other base?

A. [tex]\[ b(a) = \frac{a}{6} - 9 \][/tex]

B. [tex]\[ b(a) = \frac{a}{9} - 6 \][/tex]

C. [tex]\[ b(a) = \frac{a}{9} + 6 \][/tex]

D. [tex]\[ b(a) = \frac{a}{6} + 9 \][/tex]



Answer :

GinnyAnswer
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]

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