Understand that a function assigns each input exactly one output, use function notation to evaluate functions, and identify domain and range (TN A1.F.IF.A.1, A1.F.IF.A.2, A1.F.IF.C.5).

What this topic is asking

The Interpreting Functions standards begin with the definition. A1.F.IF.A.1: a function assigns each input exactly one output. A1.F.IF.A.2: use function notation ($f(x)$) to evaluate functions and interpret the results. A1.F.IF.C.5: identify the domain (inputs) and range (outputs), relating them to a graph. These ideas underpin the entire Functions category, the largest on the test.

What makes a relation a function

The rule is one output per input. A set of ordered pairs is a function unless some $x$-value appears twice with different $y$-values. Repeated outputs are fine: $\{(1, 5), (2, 5), (3, 5)\}$ is a function (a constant one).

The vertical line test is the graphical version: if a vertical line can cross the graph at two points, that $x$ has two outputs, so it is not a function. A parabola passes; a sideways parabola or a circle fails.

Function notation

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

Note the two directions: $g(6)$ gives an output from an input, while $g(x) = 10$ asks which input gives that output.

Domain and range

• Domain: all valid inputs ($x$-values). Read a graph left to right.
• Range: all outputs ($y$-values). Read a graph bottom to top.

For most Algebra I linear and quadratic functions the domain is all real numbers, but the range can be limited (a parabola opening up has range $y \ge$ the vertex's $y$-value). In context, the domain may be restricted (you cannot have negative time or fractional people).

How TNReady examines this topic

• Numeric response. Evaluate $f(\text{a number})$, or solve $f(x) = \text{value}$.
• Multiple choice. Identify whether a relation or graph is a function, or state the domain or range.
• Drag and drop. Match inputs to outputs from a table or mapping.

A clarifying idea is that function notation makes later topics readable: $f(2) = 5$ on a graph is the point $(2, 5)$, and the key features standard uses exactly this language to describe intercepts and extrema.

Why the "one output" rule matters

The single-output requirement is what makes a function predictable, and that predictability is why functions model the real world. If an input could map to two outputs, the model would be ambiguous: a cost function that returned both $5$ and $8$ for the same number of items would be useless. The rule also explains the vertical line test geometrically: a vertical line fixes one $x$, so crossing the graph twice would mean that one input produces two $y$-values, violating the definition. This is why a sideways parabola (which models $x$ as a function of $y$, not the reverse) is not a function of $x$. Keeping the direction straight, $f(x)$ takes an $x$ and returns one $y$, prevents most confusion in this category, especially when a problem swaps which variable is the input.

Try this

Q1. If $f(x) = 2x - 7$, find $f(5)$. [1 point]

• Cue. $f(5) = 2(5) - 7 = 3$.

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

• Cue. No: input $2$ has two different outputs ($1$ and $5$).