The function $a(b)$ relates the area of a trapezoid with a given height of 14 and one base length of 5 with the length of its other base.
It takes as input the other base value, and returns as output the area of the trapezoid.
$a(b)=14 \bullet \frac{b+5}{2}$
Which equation below represents the inverse function $b(a)$, which takes the trapezoid's area as input and returns as output the length of the other base?
A. $b(a)=\frac{a}{7}-5$
B. $b(a)=\frac{a}{7}+5$
C. $b(a)=\frac{a}{5}+7$
D. $b(a)=\frac{a}{5}-7$
Answer (2)
Start with the given function: a ( b ) = 14". f r a c b + 5 2 .
Simplify the equation: a = 7 ( b + 5 ) .
Divide both sides by 7: ". f r a c a 7 = b + 5 .
Solve for b : b ( a ) = ". f r a c a 7 − 5 . The answer is b ( a ) = ". f r a c a 7 − 5 .
Explanation
Understanding the Problem We are given the function a ( b ) = 14". f r a c b + 5 2 which represents the area of a trapezoid with height 14, one base of length 5, and another base of length b . We want to find the inverse function b ( a ) , which takes the area a as input and returns the length of the other base b .
Setting up the Equation To find the inverse function, we need to solve the equation a = 14". f r a c b + 5 2 for b in terms of a .
Simplifying the Equation First, we can simplify the equation: a = 14". f r a c b + 5 2 = 7 ( b + 5 ) a = 7 ( b + 5 )
Isolating b+5 Now, we divide both sides by 7: ". f r a c a 7 = b + 5
Finding the Inverse Function Finally, we subtract 5 from both sides to isolate b :
b = ". f r a c a 7 − 5 So, the inverse function is b ( a ) = ". f r a c a 7 − 5 .
Selecting the Correct Option Comparing our result with the given options, we see that option A matches our inverse function: b ( a ) = ". f r a c a 7 − 5 .
Final Answer Therefore, the correct answer is A.
Examples
Imagine you're designing a garden and need to calculate the length of one side of a trapezoidal flower bed. You know the area you want the flower bed to cover, the height, and the length of the other side. By using the inverse function, you can easily determine the required length of the unknown side to achieve the desired area. This is a practical application of inverse functions in everyday design and planning.
To find the inverse function b ( a ) from a ( b ) = 14 ⋅ 2 b + 5 , we simplified the equation to b = 7 a − 5 . The correct answer, matching with the given options, is A: b ( a ) = 7 a − 5 .
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