Function notation gives a precise way to describe how an output depends on an input. This guide shows how to evaluate f(x), read functions from tables and graphs, find domain restrictions, evaluate compositions, and solve equations such as f(x) = 12. Use the same workflow for homework, a calculator check, or function notation practice.
Overview
A function is a rule that assigns exactly one output to each allowed input. The notation f(x) means “the output of function f when the input is x.” It does not mean f multiplied by x.
For example, if f(x) = 3x - 2, then f(5) is found by replacing every x with 5:
f(5) = 3(5) - 2 = 15 - 2 = 13
The letter used for the input can change. If g(t) = t2 + 4, then g(3) means substitute 3 for t. The function name and the input variable identify the rule; the process remains the same.
Function notation glossary
| Term | Meaning | Example |
|---|---|---|
| Function | A rule that gives one output for each allowed input | f(x) = 2x + 1 |
| Input | The value placed into the function | x = 4 |
| Output | The result produced by the rule | f(4) = 9 |
| Domain | All permitted input values | All real numbers except 2 |
| Range | All resulting output values | y ≥ 0 for y = x2 |
| Composition | One function used inside another | f(g(x)) |
Expressions and functions are not the same
| Expression | Function notation |
|---|---|
| 2x + 5 | f(x) = 2x + 5 |
| x2 - 7 | g(x) = x2 - 7 |
| Compute by substituting into the expression | Use the named rule with a specified input |
An expression is a mathematical phrase. A function pairs that rule with an input-output relationship. This distinction matters when a problem asks you to compare f(2) and f(-2), solve f(x) = 12, or find a domain.
How to estimate
To evaluate function notation step by step, use this repeatable workflow:
- Identify the rule. Write down the formula, table, graph, or description that defines the function.
- Identify the input. In f(−3), the input is −3. In f(a + 1), the input is the entire expression a + 1.
- Substitute with parentheses. Replace the input variable everywhere it appears. Parentheses help preserve negative signs and multi-term inputs.
- Simplify in the correct order. Evaluate exponents first, then multiplication and division, then addition and subtraction.
- Check the domain. Make sure the input does not create division by zero, an even root of a negative number, or another stated restriction.
For example, let h(x) = x2 - 4x + 1. To find h(−2):
h(−2) = (−2)2 − 4(−2) + 1 = 4 + 8 + 1 = 13
Writing −22 without parentheses would be a different calculation because the exponent applies before the negative sign. Substitution parentheses prevent this common error.
Reading a table or graph
A table directly pairs inputs and outputs. If a table contains the row x = 4 and f(x) = 9, then f(4) = 9. To find an input from an output, look for the output value first. If the table shows f(−1) = 6, then the input that produces 6 is −1.
On a graph, f(3) is the y-coordinate of the function at x = 3. Move vertically from x = 3 until you reach the graph, then read the corresponding y-value. An open circle usually indicates that the point is not included; a filled point indicates that it is included. Read approximate values carefully when the graph has no exact grid intersection.
Evaluating a composition
For (f ∘ g)(x), read from the inside out: first find g(x), then use that result as the input for f. Suppose f(x) = 2x + 1 and g(x) = x2. Then:
(f ∘ g)(3) = f(g(3)) = f(9) = 2(9) + 1 = 19
In general, f(g(x)) and g(f(x)) are not automatically equal. The order of the functions is part of the problem.
Inputs and assumptions
Finding the domain means identifying which inputs are allowed. If no restriction is visible and the rule is a polynomial such as f(x) = x3 - 2x + 8, the domain is all real numbers. Other forms require a check.
- Fractions: A denominator cannot equal zero. For f(x) = 1/(x - 5), exclude x = 5.
- Even roots: The radicand must be at least zero for real-number answers. For g(x) = √(x + 2), require x + 2 ≥ 0, so x ≥ −2.
- Logarithms: The argument of a logarithm must be positive. For h(x) = log(x - 1), require x > 1.
- Context: A word problem may restrict an algebraic domain further. Time, distance, quantity, or age may need to be nonnegative or limited to whole numbers.
When a function is given as a graph, the domain is the set of x-values covered by the graph. When it is given as a table, the listed input values are the domain for that table unless the problem says the table is only a sample.
Solving for an unknown input
To solve f(x) = 12, replace f(x) with its formula and solve the resulting equation. If f(x) = 3x - 6:
3x − 6 = 12
3x = 18
x = 6
Verify by evaluating the function: f(6) = 3(6) − 6 = 12. A function can have no input, one input, or more than one input that produces a particular output, depending on its rule. For example, f(x) = x2 gives f(2) = 4 and f(−2) = 4.
Worked examples
Example 1: Linear evaluation
Let p(x) = −4x + 7. Find p(3).
Substitute 3 for x:
p(3) = −4(3) + 7 = −12 + 7 = −5
Example 2: A restricted domain
Let q(x) = √(2x − 8). Find the domain and evaluate q(6).
First require the radicand to be nonnegative:
2x − 8 ≥ 0, so x ≥ 4
The domain is [4, ∞). Now evaluate:
q(6) = √(2(6) − 8) = √4 = 2
Example 3: Solving for an input
Let r(x) = x2 − 5. Solve r(x) = 11.
x2 − 5 = 11
x2 = 16
x = 4 or x = −4
Both answers work because r(4) = 11 and r(−4) = 11.
Practice sets
Evaluate: 1. f(x) = 5x − 2, find f(−3). 2. g(t) = t2 + 2t, find g(4). 3. From a table with h(2) = −7, state the output for input 2.
Find the domain: 4. m(x) = 1/(x + 3). 5. n(x) = √(x − 9).
Solve for the input: 6. p(x) = 4x + 1, solve p(x) = 21. 7. q(x) = x2 − 1, solve q(x) = 8.
Answers: 1. −17; 2. 24; 3. −7; 4. all real numbers except −3; 5. x ≥ 9; 6. x = 5; 7. x = 3 or x = −3.
When to recalculate
Revisit a function calculation whenever the input, formula, representation, or required number system changes. A new input requires a new substitution, while a changed formula may alter both the output and the domain. Recheck especially after simplifying a rational expression, because canceling a factor does not automatically restore an input that made the original denominator zero.
Use a calculator as a verification tool rather than a replacement for setup. Enter negative and fractional inputs with parentheses, and use a graphing window that includes the relevant x-values. On many calculators, define the function in the Y= or function-entry screen, then use a table or trace feature to inspect values. Keystrokes vary by model, so confirm that the calculator interprets the expression exactly as written.
Before submitting, use this checklist:
- Did you substitute the entire input, including a negative sign or expression?
- Did you preserve the order of operations?
- Did you check denominator, radical, and logarithm restrictions?
- Does the result agree with the table or graph?
- If solving f(x) = k, did you substitute each candidate back into the function?
For related equation methods, see linear equations step by step, radical equations, and inequalities step by step. Return to this workflow whenever a new function is presented: identify the rule, substitute carefully, inspect the domain, and verify the output.
