Problem 106
Question
Describe a procedure for finding \((f \circ g)(x) .\) What is the name of this function?
Step-by-Step Solution
Verified Answer
The procedure involves evaluating the inner function \( g(x) \) at a given input then using that output as an input into the outer function \( f(x) \). The name of this operation/function is 'function composition' or 'composite function'.
1Step 1: Understand the operation
The notation \((f \circ g)(x)\) stands for the composition of two functions \( f(x) \) and \( g(x) \). 'Composition' is an operation where the output of one function, \(g(x)\), becomes the input of another function, \(f(x)\).
2Step 2: Compute the composition
To find \((f \circ g)(x)\), first evaluate the inside function at a given input, \(x\), to get \(g(x)\). Then take this result and substitute it into the other function, \(f(x)\). So, the composition of \(f\) and \(g\) is defined as \((f \circ g)(x) = f(g(x))\).
3Step 3: Identify the name of the function
The name of this function is commonly called the 'composite function' because it is made up of two other functions. Thus, we have the composite of \( f \) and \( g \), denoted \((f \circ g)(x)\).
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