Introduction
Functions describe assignments: an input is supplied, and a corresponding output is specified. Mapping diagrams give a visual way to represent these assignments. Function notation gives us a compact way to name the function and state which output goes with a particular input.
For example, in \(f(x)\), the letter \(f\) names the function and \(x\) represents its input. The full expression \(f(x)\) names the output assigned to that input. Reading these parts correctly is essential: \(f(x)\) does not mean that \(f\) and \(x\) are being multiplied.
This tutorial focuses on interpreting and writing the notation. It does not require calculating outputs from formulas. The aim is to make clear what a function statement says, both in symbols and in words.
The Parts of Function Notation
Each part has a specific role. The symbol \(f\) is the name of the function. The symbol inside the parentheses is the input. The expression \(f(x)\), taken as a whole, is the output assigned to that input.
The input does not have to be named \(x\). A function might be written \(g(t)\), \(P(n)\), or \(C(w)\). The letter inside the parentheses tells us what input is being considered; it does not change the function’s name. In \(g(t)\), for instance, \(g\) names the function and \(t\) is its input.
The letters \(x\), \(t\), and \(n\) are often called variables or placeholders. They allow us to state a rule for inputs generally, rather than naming only one particular input.
Reading a Function Statement in Words
A statement such as \(f(4)=9\) has a direct verbal reading: the function \(f\) assigns the output \(9\) to the input \(4\). Equivalently, \(9\) is the value of \(f\) at input \(4\). The number in parentheses is the input; the quantity on the right side of the equation is the output.
This reading is the same whether the output is a number, a label, or some other object. For example, if a function assigns colors to objects, a statement such as \(c(\text{apple})=\text{red}\) says that the function \(c\) assigns the label “red” to the input “apple.” Function notation records an assignment; it does not require the outputs to be numbers.
Worked Example: Reading an Assignment from a Mapping
Suppose a function \(p\) is described by the assignments below. The inputs are \(0\), \(1\), and \(2\), and their corresponding outputs are \(A\), \(B\), and \(A\).
Read the first statement by identifying its parts: \(p\) is the function name, \(0\) is the input, and \(A\) is the assigned output. In words, \(p\) assigns \(A\) to input \(0\).
For the second statement, the function name is still \(p\), the input is \(1\), and the output is \(B\). Thus, \(p(1)=B\) means that \(p\) assigns \(B\) to input \(1\).
For the third statement, \(p(2)=A\) says that \(p\) assigns \(A\) to input \(2\). The output \(A\) occurs for two different inputs here. This is consistent with the assignments: a function must give each input one output, but different inputs may share an output.
Check: Reading each statement from left to right gives the stated input and output pairs: \(0\) is assigned \(A\), \(1\) is assigned \(B\), and \(2\) is assigned \(A\). The notation records exactly the listed assignments.
Translating Verbal Rules into Notation
A verbal rule can be written in function notation by first choosing a name for the function and a symbol for its input. Then express what output the rule assigns to that input. If the rule gives an algebraic expression, that expression appears to the right of the equals sign.
For instance, suppose a function named \(h\) assigns to each input three times the input, then subtracts four. If the input is represented by \(x\), the corresponding symbolic rule is \(h(x)=3x-4\). This notation says what output the rule assigns for a general input \(x\). It does not, by itself, select a particular numerical input.
Worked Example: Writing “Twice the Input, Then Add Five”
Let \(T\) be the name of a function. The verbal rule is: “For each input, assign twice that input, then add five.” Choose \(s\) to represent a general input.
First, “twice that input” is represented by \(2s\). The words “then add five” mean that five is added to this quantity, giving \(2s+5\). Put this output expression after the function notation to state the rule:
Here \(T\) is the function name, \(s\) is the input, and \(2s+5\) describes the output assigned to that input. In words, \(T(s)=2s+5\) says that the function \(T\) assigns an output equal to twice the input plus five.
Check: The right side contains the two operations named by the rule: first twice the input, represented by \(2s\), and then an addition of five, represented by \(2s+5\). Thus the symbolic rule matches the verbal description.
The letter chosen for the input is flexible. The same rule could use \(x\) instead of \(s\), provided the notation is changed consistently: \(T(x)=2x+5\). The function name \(T\) stays the same, and the rule describes the same assignment.
Worked Example: Writing a Cost Rule with Units
Suppose a service charges a fixed fee of eight dollars, plus twelve dollars for each week of use. Let \(C\) name the cost function and let \(w\) represent the number of weeks. The variable cost is twelve dollars per week multiplied by the number of weeks, so it is represented by \(12w\) dollars. Adding the fixed fee gives the rule:
The function name \(C\) indicates that the assignment describes cost. The input \(w\) is the number of weeks, and the output \(C(w)\) is the total cost in dollars. The term \(12w\) represents the charge that depends on the number of weeks; the \(8\) represents the fixed fee.
Check: The verbal description has one fixed charge and one charge that depends on weeks. The expression \(12w+8\) includes both: \(12w\) is the weekly charge, and \(8\) is the fixed amount. The units are consistent: dollars per week multiplied by weeks gives dollars, and dollars are then added to dollars.
A Function Name Is Not Its Input
It is important to keep the function name separate from the input. In \(f(x)\), the symbol \(f\) names an assignment, while \(x\) identifies the input being supplied to it. The parentheses show that the input is being used with the function. They do not indicate multiplication.
This distinction matters because function outputs need not be numbers. A function can assign a letter or label to an input, so treating the function name as a number to multiply by the input would not even make sense in every example.
Worked Example: Distinguishing Function Application from Multiplication
Suppose a function \(q\) assigns a color to each fruit in a small set, with \(q(\text{pear})=\text{green}\). Read the notation in parts: \(q\) is the function name; “pear” is the input; and “green” is the output. Therefore, \(q(\text{pear})=\text{green}\) means that the function assigns the color green to the input “pear.”
If \(q(\text{pear})\) meant \(q\) multiplied by “pear,” the expression would require \(q\) and “pear” to be numerical factors. But in this example \(q\) names an assignment and “pear” is a label, so multiplication is not what the notation means. The function statement is instead an input-output statement.
Check: The stated assignment gives “green” as the output for “pear.” Reading \(q(\text{pear})\) as the output at that input matches the assignment. Reading it as multiplication does not describe the given function.
Why the Notation Works
The meaning of the notation follows from the idea of a function as an assignment. A function associates an output with each input in its input set. Function notation combines the name of that assignment with a particular input, so the result identifies the corresponding output.
The symbol \(f\) identifies the function under discussion.
The symbol \(x\) inside the parentheses identifies an input to which the function applies.
By the meaning of a function, that input has a corresponding output, denoted \(f(x)\).
For a concrete assignment, suppose \(f\) sends input \(2\) to output \(5\). The input-output pair is written \(f(2)=5\). This notation is not an extra operation performed on \(f\) and \(2\): it records that the output paired with input \(2\) is \(5\).
This also explains why the function name and the input symbol play different roles. The name tells us which assignment we are discussing; the input tells us where in that assignment we are looking. Changing the input can change the output, while keeping the function name fixed.
Common Mistakes and Their Corrections
A frequent mistake is to read \(f(x)\) as a product. The correction is to read the whole expression as “\(f\) of \(x\)” or “the value of \(f\) at \(x\).” The parentheses mark the input to the function; they do not mean that the function name is a numerical factor.
Another mistake is to treat the input symbol as part of the function’s name. In \(g(t)\), the function is named \(g\), not \(g(t)\). The expression \(g(t)\) denotes the output from that function for input \(t\). For a different input symbol, one could write \(g(x)\), provided \(x\) is an allowed input.
It is also easy to reverse input and output when reading an equation. In \(f(3)=7\), the input is \(3\), because it is inside the parentheses. The output is \(7\), because it is the value assigned to that input. The statement does not say that \(f(7)=3\).
| Notation | Correct reading | Common misreading |
|---|---|---|
| \(f(x)\) | The output of \(f\) for input \(x\) | The product of \(f\) and \(x\) |
| \(f(3)=7\) | \(f\) assigns output \(7\) to input \(3\) | \(f\) assigns input \(7\) to input \(3\) |
| \(g(t)\) | Function \(g\), used with input \(t\) | A function whose name is \(g(t)\) |
Why This Language Matters
Function notation lets us communicate input-output rules precisely and compactly. It makes it possible to refer to a general output, such as \(f(x)\), or to a particular assignment, such as \(f(3)=7\), without describing the entire function in a sentence each time.
Clear interpretation also prevents errors when the same function is discussed with different inputs, or when a context gives inputs and outputs units or labels. Before doing any further work with a function, ask: What is the function’s name? What is the input? What does the full expression denote? Those questions establish the meaning of the notation before any calculation begins.
For now, the essential skill is to translate accurately in both directions: read \(f(x)\) as an output assigned to input \(x\), and write a verbal input-output rule using a function name, an input symbol, and an expression for the assigned output.
Check Your Understanding
For each question, focus on the roles played by the function name, input, and output.
- In \(r(n)=12\), which symbol names the function, which represents the input, and which gives the output?
- Write a symbolic rule for a function \(A\) that assigns to each input \(x\) the input increased by nine.
- In words, what does \(D(t)=4t+6\) say about the output assigned by \(D\)?
- Explain why \(m(y)\) should not be read as the product of \(m\) and \(y\).
- A function \(k\) assigns the label “blue” to the input “square.” Write this assignment in function notation and identify its input and output.
