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

It takes as input the other base value and returns as output the area of the trapezoid.

[tex]\[ \alpha(b)=12 \cdot \frac{b+9}{2} \][/tex]

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

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

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

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

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



Answer :

GinnyAnswer
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]

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