# Introducing function notation

## Understand that a function assigns to each element in the domain exactly one element of the range.

In this section, students are introduced to function notation and begin to interpret statements that use it. They begin to build expertise in understanding how equations and inequalities that use function notation correspond to features of graphs of functions. In this section, the focus is on statements about one input and its output (i.e., one point on the graph). In section 7, they return to this correspondence, going in the opposite direction: from features of graphs to equations and inequalities about the functions represented.

1 Interpreting the graph

WHAT: Students are given a figure with two functions graphed on the same axes and asked to mark the points on the graph that illustrate statements about the two functions, e.g., $f(0) = 2$ F-IF.A.1.

WHY: This task is an opportunity for students to work with correspondences between graphs and statements that use function notation MP.1.

2 Using Function Notation I

WHAT: Students are asked whether it’s correct to simplify $C(x) = 1.25x + 2500$ by dividing both sides by $x$ and writing $C = (1.25x + 2500)/x.$

WHY: This task provides students an opportunity to grapple with a very common misconception about function notation F-IF.A.1.

3 The Customers

WHAT: Students are given a table of names and phone numbers. They are asked if two rules are functions: one assigns names to phone numbers; the other assigns phone numbers to names.

WHY: One of the main ideas for students to understand in this section is that for each input (element in the domain) there is exactly one output (element in the range) F-IF.A.1. This is achieved by having two examples for students to compare and contrast why one of the rules is a function while the other is not.

4 Points on a graph

WHAT: Students are asked two questions about a function $f$: If $10 = f(−4)$, give the coordinates of a point on the graph of $f$; if 6 is a solution of the equation $f(w) = 1$, give a point on the graph of f.

WHY: This task helps students to see the placement of the independent and dependent variables in function notation and to make sense of correspondences between symbolic expressions that use function notation and points on the graph of a function F-IF.A.1.