Problem 112 - College Algebra

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I must have made a mistake in finding the composite functions \(f \circ g\) and \(g \circ f,\) because I notice that \(f \circ g\) is not the same function as \(g \circ f\)

Step-by-Step Solution

Answer

The statement makes sense because typically, composite functions \(f \circ g\) and \(g \circ f\) are not equal.

Step 1: Understanding Composite Functions

A composite function is a function that is composed of two other functions. The composite function \(f \circ g\) is defined as \(f(g(x))\), and \(g \circ f\) is defined as \(g(f(x))\). In essence, the output of one function is used as the input for another function.

Step 2: Recognizing the Properties of Composite Functions

Though the algebraic operation of function composition might seem similar to that of multiplication or addition, composite functions do not generally obey the commutative property. That is, \(f \circ g\) is not always the same as \(g \circ f\).

Step 3: Applying the Properties to the Given Statement

From our understanding in Step 2, it is observed that the statement given makes sense. The student may not necessarily be making a mistake when they find that \(f \circ g\) is not the same function as \(g \circ f\). This is a typical characteristic of composite functions unless certain specific conditions are met.