Define a function, use function notation to evaluate, and relate domain and range to a graph and context (NC.M1.F-IF.1, F-IF.2, F-IF.5).

What this topic is asking

Three standards combine. NC.M1.F-IF.1 defines a function (each input maps to exactly one output) and its graph. NC.M1.F-IF.2 is function notation: evaluate $f(x)$ at given inputs and interpret statements like $f(3) = 7$. NC.M1.F-IF.5 relates the domain and range to a graph and a context. Together they are the language of functions used throughout the course.

What makes a relation a function

The definition is about the inputs.

A repeated x-value with different y-values breaks the rule. A repeated y-value is fine (different inputs may share an output).

Function notation

Function notation is an instruction to evaluate.

Reading the reverse direction matters too: "$f(x) = 0$" asks for the inputs that give output $0$, the zeros or x-intercepts.

Domain and range

Domain and range describe the inputs and outputs of a function.

• Domain = all valid x-values (inputs). Read left to right on a graph.
• Range = all reachable y-values (outputs). Read bottom to top on a graph.

For a line like $y = 2x + 1$, the domain and range are all real numbers. For a parabola opening up with vertex $(0, -3)$, the domain is all reals but the range is $y \ge -3$.

Domain in context

F-IF.5 stresses reasonable domains. If $x$ is the number of items bought, the domain is whole numbers $0, 1, 2, \ldots$, not all reals. If $t$ is time since launch, the domain is $t \ge 0$. Restricting the domain to fit the situation is a frequent interpretation item.

How the NC Math 1 EOC examines this topic

• Gridded response. Evaluate $f(a)$ and enter the output.
• Multiple choice. Decide whether a relation is a function, or identify the domain or range.
• Technology-enhanced. Match inputs to outputs, or select all true statements about a function's domain.

This vocabulary underpins interpreting key features and average rate of change, and it applies equally to linear, quadratic, and exponential functions.

Why the "one output" rule matters

The single-output rule is what makes a function predictable: give it an input, and you get one definite answer. That predictability is why functions model the real world so well, a cost function returns one cost for a given number of items, a height function returns one height at a given time. If an input could yield two outputs, the model would be ambiguous and useless for prediction. The vertical line test is just a visual check of this rule: a vertical line represents one input, and if it meets the graph twice, that input has two outputs. Holding the definition firmly also clarifies notation: $f(x)$ is well defined precisely because each $x$ has one $f(x)$.

Try this

Q1. If $g(x) = -2x + 6$, find $g(5)$. [1 point]

• Cue. $g(5) = -2(5) + 6 = -4$.

Q2. State the domain and range of $y = x^2$ (opening up, vertex at the origin). [2 points]

• Cue. Domain: all reals; range: $y \ge 0$.