Function notation is a compact way to name a rule and show how an input becomes an output. This cheat sheet helps students read expressions like , evaluate functions from formulas, tables, and graphs, and avoid confusing notation with multiplication. It is useful for algebra, graphing, and real-world modeling because functions describe relationships between changing quantities.
The most important idea is that means the output of function when the input is . To evaluate a function, substitute the given input everywhere the variable appears, then simplify carefully using order of operations. Students should also connect notation to tables, graphs, domain, range, and compositions such as .
Key Facts
- Function notation means the value of the function at input , not .
- To evaluate , replace every in the formula for with and simplify.
- If , then .
- The input values of a function make up the domain, and the output values make up the range.
- A relation is a function if each input has exactly one output .
- On a graph, is the -value of the point on the graph where .
- For a table, is found by locating the row where the input is and reading the matching output.
- For a composite function, means evaluate first, then use that result as the input for .
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Vocabulary
Common Mistakes to Avoid
Practice Questions
Understanding Function Notation & Evaluation Reference
A function is a rule with a responsibility. Once an allowed input is chosen, the rule must give one definite result. Different inputs may share the same result.
For example, a rule that squares a number sends both negative three and three to nine. That is still a function because neither input has been assigned two different outputs. The important restriction works in the other direction.
One input cannot split into two outputs. This explains the vertical line test on a graph.
If a vertical line hits a graph more than once, one horizontal position would have several heights. The graph does not represent a function of the horizontal variable.
Not every number is always allowed as an input. A formula can contain hidden limits. Division by zero is not defined, so any input that makes a denominator zero must be left out of the domain.
For a rule involving the square root of a number in ordinary real-number work, the quantity inside the root cannot be negative. A graph may show another kind of limit. An open circle means a point is not included, even if the curve comes very close to it.
A filled dot means the value is included. Students often make errors by treating every formula as if it accepts every number. Checking the domain before calculating can prevent a correct-looking but invalid answer.
Tables and graphs give information in different ways. A table is especially useful when data were measured at separate times or places, such as temperatures recorded each hour. It does not automatically describe values between the listed rows.
A connected graph can suggest that intermediate inputs are meaningful, but context still matters. The number of students in a class can be counted only in whole numbers. The mass of an object can vary continuously.
When reading a graph, use the scale carefully. Grid marks do not always represent one unit.
Notice whether axes begin at zero, whether the graph has breaks, and whether a point is open or filled. These details can change both the domain and the range.
Composite functions describe a chain of actions. One rule produces an intermediate result, then the next rule uses it. This is common in real situations.
A shop may first apply a discount, then calculate sales tax on the reduced price. The order matters because applying tax before the discount can give a different total. Keep the stages separate on paper, especially when parentheses, negative values, or fractions appear.
Function notation is also useful for comparing change. If a rule gives distance from time, the difference between outputs shows how far the object moved over an interval. Learning to identify the input quantity, output quantity, units, and allowed values makes function problems much easier to interpret.