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Chapter 1: Problem 70

The function \(F(x)=3 x\) converts \(x\) yards to feet. The function \(I(x)=12 x\) converts \(x\) feet to inches. Explain the meaning of \((I \circ F)(x)\).

Short Answer

Expert verified
The composite function \((I \circ F)(x)\) represents a direct conversion from yards to inches, with the formula \(36x\), where \(x\) represents the length in yards.

Step by step solution

01

Understand the Composite Function

The symbol \(\circ\) represents the composition of functions, which is the process of combining two functions where one function is performed first and the result of which is substituted in the other. In this case, \(I \circ F\) means first apply function \(F\), then apply function \(I\) to the result.
02

Define the Composite Function

Given the functions \(F(x)=3x\), which converts yards to feet, and \(I(x)=12x\), which converts feet to inches, we want to find the composite function \((I \circ F)(x)\). Apply function \(F\) first: \(F(x)=3x\) feet, this gives us the yards in feet. Then, substitute this result into function \(I\): \(I(F(x))=I(3x)=12 \cdot 3x\).
03

Simplify the Composite Function

Simplify the expression \(12 \cdot 3x\) to yield \(36x\). Therefore, the composite function \((I \circ F)(x)\) is \(36x\).

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