Determine whether a relation is a function from a table, graph, mapping, or equation, and use and evaluate function notation f(x) (A.F.1).

What this topic is asking

A.F.1 asks you to decide whether a relation is a function (from a table, graph, mapping, or equation) and to use and evaluate function notation $f(x)$. On the Virginia Algebra I SOL these are Functions items: identify a function, evaluate $f$ at a value, or interpret what $f(a) = b$ means. They appear as multiple choice, fill-in-the-blank (type the value), and drag-and-drop (match inputs to outputs).

What makes a relation a function

A relation is any set of input-output pairs. It is a function when each input has exactly one output. Inputs are the $x$-values (the domain); outputs are the $y$-values (the range). The defining rule is one-way: an input cannot point to two different outputs, but two different inputs may share the same output.

So $\{(1, 5), (2, 5), (3, 5)\}$ is a function (three inputs, each with one output, even though the output repeats), but $\{(1, 5), (1, 6)\}$ is not (the input $1$ has two outputs).

The vertical line test

For a graph, use the vertical line test: if any vertical line crosses the graph in more than one point, the graph is not a function, because that vertical line marks a single $x$ with multiple $y$-values. A straight line, a parabola opening up or down, and an exponential curve all pass. A circle, a sideways parabola, and a vertical line all fail.

Function notation

Function notation writes the output as $f(x)$, read "$f$ of $x$." The letter $f$ names the function and $x$ is the input. So $f(x) = 2x + 1$ is a rule: take an input, double it, add one. The notation $f(3)$ means the output when the input is $3$.

Interpreting function notation

The equation $f(a) = b$ packs two facts: the input is $a$ and the output is $b$, which is the ordered pair $(a, b)$. SOL items use this in context. If $C(t)$ is the cost after $t$ months, then $C(6) = 200$ means "after $6$ months the cost is \200$." Reading the input and output correctly, rather than computing blindly, is what these items reward. You may also be asked to **solve**$f(x) = b$: find the input(s) that give a known output, which reverses the evaluation.

Why the one-output rule defines a function

The single-output requirement is what makes a function predictable, and that is the whole reason the concept exists. A function is a rule you can rely on: give it an input and it returns one definite output, every time. If an input were allowed two outputs, the rule would be ambiguous, you could not say what $f(3)$ "is," because it would be both values at once. That ambiguity is exactly why a vertical line, or a relation with a repeated input, is not a function: at that input the rule cannot decide. Outputs are allowed to repeat (many inputs can map to the same output) because that does not create ambiguity, asking "what does input $2$ give?" still has one answer even if input $5$ gives the same thing. This is the foundation for everything else in the Functions strand: domain and range, graphing, and comparing families all assume each input has a single, well-defined output.

How the SOL examines this topic

• Multiple choice. Identify which relation, table, or graph is a function; pick the value of $f(a)$.
• Fill-in-the-blank. Evaluate $f$ at a given input and type the result.
• Drag-and-drop. Match inputs to outputs, or complete a function table.

Try this

Q1. For $g(x) = 2x - 7$, find $g(4)$. [1 point]

• Cue. $2(4) - 7 = 1$.

Q2. Is $\{(2, 3), (2, 5), (4, 1)\}$ a function? [1 point]

• Cue. No: input $2$ has two outputs.