Use function notation to evaluate and interpret functions, decide whether a relation is a function, and identify domain and range from equations, tables, and graphs (Ohio F-IF.1, F-IF.2, F-IF.5).

What this topic is asking

Ohio standards F-IF.1 and F-IF.2 ask you to understand what a function is and to use function notation. A function pairs each input with exactly one output. The notation $f(x)$ names the output the function gives for input $x$. F-IF.5 adds domain (the allowed inputs) and range (the resulting outputs). Functions are the largest reporting category, so this vocabulary underlies much of the test.

What makes a relation a function

A relation is any set of input-output pairs. It is a function only if every input has one and only one output.

A line $y = 2x + 1$ is a function; a circle and a vertical line are not.

Function notation: evaluating and solving

The symbol $f(x)$ is read "$f$ of $x$" and stands for the output, not multiplication.

Domain and range

The domain is the complete set of inputs; the range is the complete set of outputs.

For a graph, sweep left to right for the domain and bottom to top for the range. A line extends forever, so its domain and range are all real numbers. A context can restrict the domain: if $x$ is a number of items, the domain is whole numbers $\geq 0$.

How Ohio examines this topic

• Numeric and equation response. Evaluate $f(a)$, or solve $f(x) = k$ for the input.
• Multiple choice and multiple-select. Decide whether a relation, table, or graph is a function; choose the domain or range.
• Tables and drag and drop. Complete a function table, or match inputs to outputs.

Because some items are exact-match, read carefully whether you are asked for an output ($f(a)$) or an input ($f(x) = k$).

Why each input gets exactly one output

The "one output per input" rule is what makes a function predictable: feed in a value and you always get back a single, definite result. If an input could give two outputs, the rule would be ambiguous, and notation like $f(4)$ would not name one number. This is precisely what the vertical line test detects: a vertical line is the set of points sharing one $x$, so two crossings mean that one input produced two outputs, breaking the rule. Outputs, by contrast, are free to repeat, many inputs can lead to the same output, which is why a horizontal line still passes. Holding onto this asymmetry keeps domain (inputs) and range (outputs) straight.

Why domain can be restricted

The domain is not always "all real numbers." A formula can forbid certain inputs (for Algebra I, mainly dividing by zero or taking an even root of a negative), and a context can restrict it further. If $C(n)$ gives the cost of $n$ tickets, then $n$ must be a whole number $\geq 0$, no one buys $-2$ or $2.5$ tickets, so the realistic domain is $\{0, 1, 2, \ldots\}$ even though the formula would accept any number. Matching the domain to the situation is part of interpreting a model, and the test often asks for the reasonable domain rather than the formula's natural domain.

Try this

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

• Cue. $f(-1) = 5 - 2(-1) = 7$.

Q2. For $g(x) = 4x$, solve $g(x) = 20$. [1 point]

• Cue. $4x = 20$, so $x = 5$.