Evaluate and interpret function notation, determine whether a relation is a function, and identify the domain and range of a function from multiple representations (MA.912.F.1.1, MA.912.F.1.2).

What this topic is asking

MA.912.F.1 introduces the language of functions. You evaluate function notation like $f(x)$, decide whether a relation is a function (each input has exactly one output), and state the domain (inputs) and range (outputs) from a graph, a table, or a context. On the B.E.S.T. Algebra 1 EOC these are everywhere, because almost every later topic is phrased in function language.

Function notation

$f(x)$ is read "f of x" and names the output of the function $f$ for the input $x$. To evaluate, substitute:

You can also work backward: solving $f(x) = 11$ for $2x + 1 = 11$ gives $x = 5$. The notation is not multiplication, $f(3)$ is a single output value, not "f times 3."

Is it a function?

A relation is a function if each input maps to exactly one output. A table is a function unless an $x$-value repeats with different $y$-values. On a graph, use the vertical line test: if any vertical line crosses the graph more than once, it is not a function (that $x$ has two outputs). A circle fails; a parabola opening up passes.

Domain and range in context

In a real situation, the domain and range are limited by what makes sense. For a function giving ticket revenue from the number of tickets sold, the domain is whole numbers from $0$ up (you cannot sell $-3$ or $2.5$ tickets), which makes it discrete. For a function giving distance over continuous time, the domain is an interval like $0 \le t \le 5$, which is continuous. The EOC rewards matching the domain and range to the context, not defaulting to "all real numbers."

How the B.E.S.T. EOC examines this topic

• Multiple choice and equation editor. Evaluate $f(a)$, or solve $f(x) = b$.
• GRID and multiselect. Identify the domain or range from a graph, or select all relations that are functions.
• Context items. State a reasonable domain and range and interpret them.

A clarifying idea: domain and range are just the shadows of the graph. The domain is what the graph covers if you flatten it onto the $x$-axis, and the range is what it covers flattened onto the $y$-axis. Picturing those two shadows makes reading them off a graph automatic.

Why the vertical line test works

The vertical line test is not a separate rule, it is the definition of a function drawn as a picture. A vertical line is the set of all points with a single $x$-value. If that line meets the graph at two points, then that one input $x$ has produced two different outputs $y$, which is exactly what a function forbids. So the test simply checks the defining condition (one output per input) at every input at once. This is why a sideways parabola or a circle is not a function: a single $x$ near the middle corresponds to both a top and a bottom $y$. Understanding the test this way means you never have to memorize it, you can reconstruct it from what a function is.

Discrete versus continuous, and why it matters

Whether a domain is discrete or continuous changes how you report and graph it. A discrete function (tickets sold, students enrolled) has a domain of separate points, so its graph is dots, not a connected curve, and its domain is listed or restricted to integers. A continuous function (height over time, temperature) has a domain that is a full interval, drawn as an unbroken curve. On the EOC, a context item may ask you to choose between a connected line and a set of points, and the right choice follows from whether the input can take in-between values. Counting situations are discrete; measuring situations are usually continuous.

Try this

Q1. If $g(x) = x^2 - 4$, find $g(-3)$. [1 point]

• Cue. $g(-3) = (-3)^2 - 4 = 9 - 4 = 5$.

Q2. A parking garage charges by the whole hour. Is the cost function discrete or continuous? [1 point]

• Cue. Discrete, because hours are counted in whole-number steps.