Problem 75 - College Algebra

Question

Describe a procedure for finding \((f \circ g)(x) .\) What is the name of this function?

Step-by-Step Solution

Answer

The composition of function \(f\) and function \(g\), in the form \((f \circ g)(x)\), means first applying function \(g\) to \(x\), and then applying function \(f\) to the result. The name of this function is called the composition of \(f\) and \(g\).

Step 1: Understand Function Composition

Function composition is an operation that takes two functions \(f\) and \(g\) and produces a function \(h = f \circ g\). In other words, \(h(x) = f(g(x))\). It is called function composition because it composes the two functions \(f\) and \(g\) to create a new function \(h\).

Step 2: Apply Function Composition

To find \((f \circ g)(x)\), substitute \(x\) into function \(g\) first, and then substitute the result into function \(f\). This means applying the rule of function \(f\) to the output of function \(g\).

Step 3: Simplify

After substitution, simplify the expression to find the definite form of \((f \circ g)(x)\).