**Questions Eliciting Thinking**What is a function?
Can you explain why you thought this sequence was/was not a function?
Could you also graph your second table of values? Is the ability to be graphed what makes this a function?
What did you mean by the *x-*values cannot repeat?
If every input had the same output, would that be a function?
Are all functions linear? |

**Instructional Implications**Review the definitions of *relation* and *function*Â emphasizing that a function is a relation in which every input value is paired with only one output value. Provide examples of relations that are functions and relations that are not functions described in a variety of ways (e.g., tables of values, mapping diagrams, algebraic rules, graphs, verbal descriptions). Be sure to include many nonlinear examples of functions. Guide the student to carefully consider each example to determine whether or not it represents a function. Model explaining and justifying the reasoning behind the determination.
Explain the difference between defining the concept of a *function* and a test used to detect functions such as identifying repeated values of *x* in a table or using the vertical line test. Explain the rationale behind the vertical line test by directly relating it to the definition of a function. Be sure the student understands that if a vertical line intersects a graph in more than one point, each of the points of intersection contain the same *x*-coordinate but different *y*-coordinates. Consequently, the same value of *x* has been paired with more than one value of *y,* so the graph cannot represent a function. Present the student with additional examples and non-examples of graphed functions. Expose the student to a variety of graphs including linear, quadratic, cubic, rational, absolute value, exponential, step, and piecewise. Include both horizontal and vertical lines. Have the student indicate whether or not each graph represents a function and justify his or her answers.
Consider implementing the MFAS task *Identifying Functions* (F-IF.1.1).
Review the concept of a sequence and provide examples of a variety of arithmetic, geometric, and other types of sequences. Explain that the terms of a sequence comprise the range of a function while their term numbers comprise the domain. Guide the student to observe that since each term has a unique term number, sequences are examples of functions. Emphasize that the domain of a sequence is a subset of the integers. Provide additional examples of sequences, and ask the student to explain why each is a function and to describe both the domain and range. |

**Instructional Implications**Explain that the terms of a sequence comprise the range of a function while their term numbers comprise the domain. Illustrate this by asking the student to make a table of values for each sequence. Emphasize that the domain of a sequence is always a subset of the integers. Model a verbal description of each domain (e.g., â€śThe domain is the set of integers greater than or equal to oneâ€ť). Also show the student how to use notation to write each domain (e.g., {1, 2, 3, 4, â€¦} or {*x:x *Â Integers and *xÂ *= 1}).
If needed, assist the student in improving his or her explanation for why each sequence is a function. Provide additional examples of sequences, and ask the student to explain why each is a function and to describe both the domain and the range. |