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A1.F.1.2 Identify the dependent and independent variables as well as the domain and range given a function, equation, or graph. Identify restrictions on the domain and range in real-world contexts.

In a Nutshell

Students will understand f is a function and x is an element of its domain, and f(x) denotes the output of f corresponding to the input x. The f(x) is the dependent variable, as it changes with x, the independent variable. Restrictions may occur when x- and/or y-values do not exist in a real-world context. When finding the domain, the denominator of a fraction cannot be zero and the number under a square root sign must be a positive value. 

Student Actions

Teacher Actions

  • Students will examine functions to determine what happens to the output (y value) when the input (x value) changes a specific amount.
  • Students will develop a deep and flexible conceptual understanding in both algebraic and real-world contexts when specific input or output values are not possible.
  • Students have a productive mathematical disposition when asked to identify the x- and /or y-values that do not exist in a real-world situation

  •  Students will  interpret and translate functions both verbally and graphically and communicate mathematically about the variables and their restrictions.


  • Provide mathematical representations of functions and allow the students to make connections between the domain(x-values) and how it affects the range(y-values).

  • Facilitate meaningful mathematical discourse while students discuss reasons to determine the dependent and independent variables as expressed in the real-world situation, or in a graph.

  • Pose purposeful questions supporting student exploration of  domain, range and their restrictions.

  • Elicit and use evidence of student thinking as  students explain their understanding of variables and how to identify restrictions.

  • Support productive struggle as students work through determining the independent and dependent variables in real-world situations.


Key Understandings


  • Understand domain as the set of x-coordinates within a  set of points on a graph or within a  written set of ordered pairs.  These values are the input to a function or relation. 

  • Understand the range as the y-coordinates within points on a graph or within a written set of ordered pairs. These value are the output of a function or relation. 

  • Identify the independent variable as the variable whose value determines the value of other variables. Example: In the formula for the area of a circle, A = πr2, r is the independent variable, as its value determines the value of the area (A). 

  • Identify the dependent variable as the  variable whose value is determined by the value of an independent variable.Example: In the formula for the area of a circle, A =πr2, A is the dependent variable, as its value depends on the value of the radius (r).



Using the function y = 3x - 7, x is usually defined as the independent variable and the value of y is typically dependent on the choice of the value of x).



Understand that there are restrictions on domain and range in algebraic situations as well as real-world situations.


y=x2 The range must be greater than or = 0.



Real world situations like the amount of money someone who is paid by the hour makes in a day depends on the number of hours he or she works during that day. In this case the amount of money being paid (independent variable) depends on the time worked (dependent variable).


The independent variable and the domain are the x-values of the Total Sales. The dependent variable and the range are the y-values of the Total Pay. The domain is x>0, which is restricted to positive real numbers and the range is y>75, which is restricted to positive real numbers greater than or equal to 75.


Total daily pay of a salesperson
  • Students misidentify  the dependent and independent variables.

  • Students cannot identify the restrictions in a real-world situation.

  • Students confuse the domain and range restrictions.

  • Students struggle with determining the domain and range.


OKMath Framework Introduction

Algebra 1 Introduction


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