Chapter 1: The Independent Variable and Function Notation

Functions and Independent Variables

• Definition of Variable Roles:

  • An independent variable serves as the input value to a function, determining the resulting output value of the dependent variable.
  • In the mathematical relationship where an expression equals $y^2$ to define $x$, the variable $y$ is designated as the independent variable.

• Functional Notation with $y$ as the Input:

  • When $y$ is established as the independent variable, the equation expressing $x$ in terms of $y$ is represented using functional notation such as $h(y)$.
  • The complete functional relationship is defined as:         $h(y) = y^2$
  • In this notation, $x = h(y)$, emphasizing that $x$ depends on the values assigned to $y$.

Converting Functional Perspectives

• Expressing $y$ as a Function of $x$:
  • Standard algebraic convention often reorient relationships to express $y$ explicitly as a function of $x$ (i.e., starting with "$y =$").
  • Rearranging $x = y^2$ to solve for $y$ yields:         $y = \pm \sqrt{x}$
  • This conversion changes the variable roles, making $x$ the independent variable and $y$ the dependent variable.