Determine whether a relation is a function, use function notation to evaluate and interpret functions, and identify domain and range from graphs and tables (A.FGR, Functional and Graphical Reasoning).

What this topic is asking

This Functional and Graphical Reasoning (A.FGR) standard establishes the language of functions that the entire Functions domain (about 30 percent of the EOC) relies on. It asks three things: decide whether a relation is a function, use function notation to evaluate and interpret, and find domain and range. The Georgia Milestones EOC tests these as quick selected-response items (evaluate $f(4)$, identify domain) and as short justification items (is this relation a function and why). Getting comfortable with $f(x)$ notation early pays off everywhere, because every linear, exponential, and quadratic function is written this way.

What makes a relation a function

A relation pairs inputs with outputs. It is a function when every input has exactly one output. Inputs may share an output (two $x$-values can map to the same $y$), but a single input may not have two outputs.

• The set $\{(1, 2), (3, 4), (5, 8)\}$ is a function: each input appears once.
• The set $\{(1, 2), (1, 6), (3, 4)\}$ is not: the input 1 has two outputs.

Function notation

Function notation $f(x)$ reads "f of x" and names the output when the input is $x$. It is not the product of $f$ and $x$.

• To evaluate, substitute: for $f(x) = 3x - 5$, $f(4) = 3(4) - 5 = 7$.
• To interpret, read units: if $C(t)$ is cost after $t$ months, then $C(6) = 130$ means the cost after 6 months is $130.
• To solve $f(x) = 0$, set the rule equal to 0 and solve for $x$ (this gives the x-intercepts).

Domain and range

The domain is the set of permissible inputs ($x$-values); the range is the set of resulting outputs ($y$-values).

• From a graph, read domain left-to-right (the spread of $x$) and range bottom-to-top (the spread of $y$).
• From a table or set of points, the domain is the list of inputs and the range is the list of outputs.
• In a context, restrict to sensible values: a number of items is a whole number $\ge 0$, time is usually $\ge 0$.

How the Milestones examines this topic

• Multiple choice. Evaluate $f$ at a value, or pick the domain or range of a graph or table.
• Drag and drop. Sort relations into "function" and "not a function," or match inputs to outputs.
• Constructed response. Justify whether a relation is a function, or interpret a function value in context.

Why "one output per input" is the whole idea

Every later property of functions traces back to the single rule that each input has exactly one output. It is what makes function notation unambiguous: $f(4)$ names one number, so you can compute with it and compare it. It is what the vertical line test detects on a graph, because a vertical line is the set of all points sharing one input, and a function can meet it only once. And it is what lets a function be inverted only when outputs are also unique. Internalizing the rule, rather than memorizing the vertical line test as a separate fact, means you can decide "function or not" from a graph, a table, a mapping diagram, or a set of points using the one definition, which is exactly the flexibility the EOC items demand.

Reading domain and range in context

Context often narrows the domain and range below what the algebra alone allows. The rule $h(t) = 5 + 2t$ has all real numbers as its mathematical domain, but as a model of a plant's height it only makes sense for $t \ge 0$ (you cannot have negative weeks), and the range starts at 5 (the height when planting) and grows. On the EOC, a domain or range question attached to a word problem is really asking you to apply this common-sense restriction, so read the units and ask what inputs the situation permits before stating the answer.

Try this

Q1. If $g(x) = x^2 - 1$, find $g(3)$. [1 point]

• Cue. $g(3) = 3^2 - 1 = 8$.

Q2. Is $\{(2, 5), (4, 5), (6, 9)\}$ a function? [1 point]

• Cue. Yes; each input has one output (sharing the output 5 is allowed).