Problem 51 - College Algebra

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When all is said and done, it seems to me that direct variation equations are special kinds of linear functions and inverse variation equations are special kinds of rational functions.

Step-by-Step Solution

Answer

Yes, the statement makes sense because it correctly classifies direct variation equations as a special type of linear functions (since they create a straight line when graphed) and inverse variation equations as a special type of rational functions (since they create a hyperbola when graphed).

Step 1: Understanding Direct Variation Equations

Direct variation equations are a specific type of linear functions where the equation is in the form \(y = kx\), where 'k' is the constant of variation. In other words, as 'x' increases, 'y' also increases directly or as 'x' decreases, 'y' decreases. This means that the direct variation equation forms a straight line when graphed, characterizing it as a linear function.

Step 2: Understanding Inverse Variation Equations

Inverse variation equations are equations of the form \(y = k/x\), where 'k' is the constant of variation. As the name suggests, 'y' varies inversely as 'x'. So, as 'x' increases, 'y' decreases and vice versa. These equations produce a hyperbola when graphed, which categorizes them as rational functions, as rational functions are any functions that can be expressed as the quotient of two polynomial expressions.

Step 3: Validating the Statement

The given statement says, 'Direct variation equations are special kinds of linear functions and inverse variation equations are special kinds of rational functions.' Based on the definitions elaborated in the previous steps, it is clear that the statement does make sense as it correctly categorizes direct variation equations under linear functions and inverse variation equations under rational functions.