Function Notation and Input-Output Tables

Function notation labels an input-output relationship. The symbol f(x) is read 'f of x' and represents a value, not a multiplication sign.

Evaluate a function

If f(x) = 2x² − 3, then f(4) means substitute 4 for x: 2(4²) − 3 = 32 − 3 = 29. Use parentheses around negative inputs so the exponent applies correctly: f(−2) = 2(−2)² − 3 = 5.

Complete an input-output table

For the rule g(x) = x + 5, an input of 0 gives g(0) = 5, an input of 2 gives 7, and an input of −3 gives 2. Each row uses the same rule. If a table includes an output but not its input, solve the equation backward to find the missing input.

Check domain restrictions

Not every input is allowed for every function. For a rational function, a denominator cannot be zero; for a real-valued square root, the radicand generally must be nonnegative. A table should include only inputs in the function's domain unless the problem defines another convention.

Common mistakes

• Treating f(3) as f times 3 instead of an output at input 3.
• Substituting into only one occurrence of x.
• Forgetting parentheses around a negative value before squaring.
• Using a table pattern without checking it against the stated rule.
• Ignoring a domain restriction that makes an input undefined.

Apply the order of operations during substitution

Substitute the entire input before simplifying. For f(x) = 3x² − 2x, f(−2) = 3(−2)² − 2(−2) = 12 + 4 = 16. Without parentheses, it is easy to square only part of the input or lose the negative sign. If the rule is f(x + 1), the expression inside the parentheses is the input; f(3) means replace every x by 3, producing f(3) = (3 + 1) under that rule.

Understand a table as ordered pairs

Each row of an input-output table represents an ordered pair (input, output). For a function, the same input cannot map to two different outputs. If a table lists x = 2 with outputs 5 and 7, it does not represent a function of x unless the context defines some additional feature that distinguishes the rows. Different inputs can share the same output; that is allowed. A function need not be one-to-one.

Read function notation for transformed inputs

f(a) is the output when input a is used. It is not the product f times a. Expressions such as f(x + 2), f(2x), and f(x) + 2 mean different operations: substitute x + 2 as the input, substitute 2x as the input, or add 2 to the output. For f(x) = x², f(x + 2) = (x + 2)² = x² + 4x + 4, while f(x) + 2 = x² + 2. Keep the input transformation inside the function separate from changes made afterward.

Find an input from an output

If a function rule gives an output and asks for the input, set the rule equal to that output and solve. For g(x) = 3x + 1, an output of 16 means solve 3x + 1 = 16, so x = 5. Check the candidate by evaluating g(5). If multiple inputs produce the same output, report all allowed inputs or use the domain information supplied.

Check domain and table patterns

An input must be in the function's domain. A rational rule cannot use an input that makes its denominator zero; a real square-root rule generally requires a nonnegative radicand. A finite table may suggest a pattern but does not always determine a unique rule. Use the function definition or stated context rather than assuming the pattern continues beyond the listed inputs.

A reliable evaluation routine

1. Identify exactly what expression is the input.
2. Substitute it for every variable occurrence using parentheses.
3. Simplify in the correct order and preserve signs.
4. Check that the input is in the domain.
5. For a table, confirm that each row follows the same rule and that no input has conflicting outputs.

Read f(x) as the output for a specific input

Function notation names a rule: f(x) is the output assigned to input x. It does not mean f multiplied by x. If f(x) = 3x − 2, then f(5) = 3(5) − 2 = 13. If the input is negative, use parentheses: f(−2) = 3(−2) − 2 = −8. Follow order of operations after substitution.

A function gives exactly one output for each permitted input. A table violates this rule if the same input appears with two different outputs. Repeated outputs are allowed: different inputs may map to the same output. On a graph, the vertical-line test checks that each x-value corresponds to at most one y-value.

Use an input-output table to infer a rule

Look at how outputs change when inputs change by equal amounts. If every one-unit increase in x adds 4 to y, a linear rule with slope 4 is possible. Use a known pair to find the starting value: if y = 4x + b and the table includes (2, 11), then 11 = 8 + b, so b = 3. Test the proposed rule against the other rows.

Not every function is linear. If first differences change but second differences are constant, a quadratic rule may fit; if outputs multiply by a fixed ratio for equal input steps, an exponential pattern may fit. A small table can be consistent with multiple rules, so use any stated domain or expected model and avoid claiming a unique formula without enough information.

Understand domain, range, and inverse questions

The domain is the set of allowed inputs; the range is the set of resulting outputs. A context can restrict inputs to whole numbers, nonnegative values, or a specific interval even if the formula works more broadly. For f(x) = 1/(x − 2), x = 2 is excluded because it makes the denominator zero. For f(x) = √x in real numbers, negative inputs are excluded.

A question asking for the input that gives an output sets f(x) equal to that output and solves. If f(x) = 2x + 1 and the output is 9, solve 2x + 1 = 9, giving x = 4. Check the result in the original function and ensure it belongs to the domain.

• Substitute the input into the expression for f and use parentheses as needed.
• Check that each input has exactly one output.
• Use tables to test a rule across all provided pairs.
• Identify domain restrictions from context and the formula.
• Solve f(x) = target to find an input, then verify it.

Exam takeaway

Function notation identifies an output for a given input. Substitute the input into every x in the rule, simplify carefully and verify that the input is in the domain.