Understand function notation and evaluate functions, and determine the domain and range from a rule, a graph, or a table (LA A1: F-IF.A.1, F-IF.A.2, F-IF.B.5).

What this topic is asking

Standards A1: F-IF.A.1, F-IF.A.2, and F-IF.B.5 ask you to understand function notation, evaluate a function, and find its domain and range from a rule, a graph, or a table. On LEAP 2025 these are Type I Major Content items, and function notation underpins almost every other functions topic. Evaluating $f(x)$ is also a no-calculator skill for Session 1a.

Function notation and evaluating

$f(x)$ is read "$f$ of $x$" and names the output of the function $f$ at input $x$. To evaluate, substitute the given number for every $x$ and simplify.

The notation does not mean $f$ times $x$; it is a single symbol naming the output.

The meaning of a function

A relation is a function when each input has exactly one output. On a graph, this is the vertical-line test: if any vertical line hits the graph more than once, the relation fails (one $x$ would have two $y$'s). In a table, a function never repeats an $x$ with two different $y$'s.

Domain and range

From a table or a list of points, the domain is the set of $x$'s and the range is the set of $y$'s. From a graph, scan left to right for the domain and bottom to top for the range. From a rule, the domain is all real numbers unless an operation restricts it (you cannot divide by zero or take an even root of a negative).

How LEAP examines this topic

• Equation response. Evaluate $f(a)$ for a given input.
• Multiple choice. Identify the domain or range from a list, table, or graph; or apply the vertical-line test.
• Drag and drop. Match inputs to outputs, or sort relations into function and not-a-function.

A clarifying idea: in a context, the domain may be restricted by reality. If $f(t)$ models the height of a ball over time, negative $t$ has no meaning, so the practical domain starts at $t = 0$.

Why each input has exactly one output

The "one output per input" rule is what makes a function predictable, and it is the conceptual core of F-IF.A.1. A function is a dependable machine: feed it a value and it returns a single, determined result, so $f(3)$ has one and only one meaning. If a single input could produce two outputs, the notation $f(3)$ would be ambiguous, you could not say what "the output" is, and you could not graph, model, or solve reliably. This is exactly what the vertical-line test detects: a vertical line fixes one $x$, and the line meeting the graph twice would mean that one $x$ has two $y$'s, breaking the rule. Note the rule is one-directional: different inputs are allowed to share an output (a horizontal line may hit the graph many times), which is why range values can repeat but domain values cannot. Understanding functions as single-valued machines is what lets later topics talk confidently about "the" value, "the" rate of change, and "the" maximum of a function.

Try this

Q1. For $f(x) = 5 - 2x$, find $f(-1)$. [2 points]

• Cue. $f(-1) = 5 - 2(-1) = 5 + 2 = 7$.

Q2. A function has points $\{(0, 3), (2, 3), (4, 5)\}$. What is its range? [1 point]

• Cue. Range is the set of outputs: $\{3, 5\}$.