Functions: Key Concepts, Examples, and Study Guide
In algebra, a function assigns exactly one output to every allowed input. Test a relation by checking whether any input has two different outputs, read f(x) as the output of function f at input x, and respect the domain before calculating. Then verify the result with another representation, such as a table or graph.
What Is a Function in Algebra?
A relation connects inputs and outputs. It qualifies as a function when every input in its domain has exactly one output. Different inputs are allowed to produce the same output, but one input cannot produce two different outputs.
For example, the relation {(1,4), (2,5), (3,4)} is a function. The output 4 appears twice, but each input appears with only one output. The relation {(1,4), (1,6), (2,5)} is not a function because the input 1 is paired with both 4 and 6.
OpenStax defines the domain as the set of inputs and the range as the set of resulting outputs. This input-output condition works for numbers, names, objects, and other mathematical sets, although algebra courses usually use numerical inputs and outputs.
How to Tell Whether a Relation Is a Function
The same test works in every representation: look for an input with more than one output.
Representation | What to inspect | Function test |
Ordered pairs | First coordinate in each pair | A repeated input fails only when its outputs differ |
Table | Input column | Each input row must lead to one output value |
Mapping diagram | Arrows leaving each input | Each input must have exactly one outgoing arrow |
Graph | Vertical lines | No vertical line may intersect the graph more than once |
Formula or situation | Meaning of the rule | Every allowed input must determine one output |
The visual below compares the same condition in two mapping diagrams.
A vertical line test applies when x is the horizontal input. If a vertical line crosses the graph twice, that x-value has two outputs, so the graph does not represent y as a function of x. A horizontal line is used for a different question: whether a function is one-to-one.
Function Notation and Evaluation
The notation f(x) identifies the function and its output. The letter f is the function's name, x is the input, and f(x) is the output produced by that input. The parentheses indicate the input; they do not mean multiplication.
Suppose f(x)=2x²-3 and you need f(-2).
- Replace every x with (-2): f(-2)=2(-2)²-3.
- Evaluate the exponent first: (-2)²=4.
- Finish the arithmetic: 2(4)-3=5.
- State the result: f(-2)=5.
Keeping parentheses around a negative or multi-term input prevents sign and distribution errors. For f(x)=x²+1, the expression f(a+1) becomes (a+1)²+1, not a+1²+1.
Function notation can also be read backward. If f(x)=3x-2 and f(x)=10, solve 3x-2=10 to find the input x=4. Evaluating a function finds an output from an input; solving a function equation finds input values that produce a specified output.
Four Ways to Represent a Function
A function may appear as a situation, table, rule, or graph. The representation changes, but the input-output relationship must remain the same.
Imagine a school event charges a $2 entry fee plus $3 per activity. Let n be the number of activities and C(n) the total cost.
- Situation: start with $2, then add $3 for each activity.
- Rule: C(n)=3n+2.
- Table: (0,2), (1,5), (2,8), (3,11).
- Graph: plot the table values; they lie on a line with vertical intercept 2 and rate of change 3.
Converting between forms also checks the work. The rule gives C(2)=8, so the table must contain (2,8) and the graph must pass through that point. If one representation disagrees, revisit the translation before continuing.
Domain and Range
The domain is the set of allowed inputs. The range is the set of outputs the function actually produces. A formula can impose restrictions, and the situation can add more.
For real-number algebra, check restrictions in this order:
- Check denominators. A denominator cannot equal zero. For r(x)=1/(x-4), exclude x=4.
- Check even roots. The quantity under an even root must be nonnegative. For g(x)=√(x+3), require x+3≥0, so x≥-3.
- Check the context. A time, item count, or physical length may need to be nonnegative or whole-valued even when the formula accepts more inputs.
- Find the resulting outputs. Use the rule, table, or graph to determine the range; do not assume it matches the domain.
The OpenStax domain and range section applies the same restrictions to formulas and graphs. In advanced courses, complex-number domains can change the even-root restriction, so state the number system when it matters.
Worked Examples and Checks
Example 1: Test a table
Does the table represent a function?
x | y |
-2 | 5 |
0 | 1 |
3 | 5 |
3 | 7 |
The input 3 produces both 5 and 7, so the relation is not a function. The repeated output 5 is acceptable; the repeated input with different outputs is the failure.
Example 2: Evaluate and verify
Let h(t)=t²-4t+7. Find h(3).
h(3)=3²-4(3)+7=9-12+7=4.
Check by substituting again in a different grouping: h(3)=(3-2)²+3=1+3=4. The two equivalent forms agree. You can also plot the point (3,4) and confirm that it lies on the graph of h.
Example 3: Find a domain
Find the real-number domain of q(x)=√(2x-6)/(x-5).
- Even-root condition: 2x-6≥0, so x≥3.
- Denominator condition: x≠5.
- Domain: [3,5)∪(5,∞).
Check the boundaries: x=3 makes the square root zero and is allowed; x=5 makes the denominator zero and is excluded.
Major Function Families
Recognize a family by its rule and graph behavior. The list below is a starting map; always check the actual formula.
Family | Typical form | Visual clue | Common domain note |
Constant | f(x)=c | Horizontal line | Usually all real numbers |
Linear | f(x)=mx+b | Straight nonvertical line | Usually all real numbers |
Quadratic | f(x)=ax²+bx+c | Parabola | Usually all real numbers |
Absolute value | f(x)=|x| | V-shape | Usually all real numbers |
Polynomial | f(x)=aₙxⁿ+⋯+a₀ | Smooth curve without breaks | Usually all real numbers |
Rational | f(x)=p(x)/q(x) | May have breaks or asymptotes | Exclude zeros of q(x) |
Radical | f(x)=√x or related form | Often begins at an endpoint | Check even-root radicands |
Exponential | f(x)=abˣ | Changes by a multiplier | Base conditions matter |
A linear equation such as 3x+2=14 asks for values that make one statement true. A linear function such as f(x)=3x+2 describes an output for every allowed input. Use the linear equation guide when the task is solving; use function tools when the task is evaluating, graphing, finding domain or comparing behavior.
For an exponential function, the input appears in the exponent. Review the exponent rules before evaluating, simplifying, or interpreting exponential expressions.
Common Function Mistakes
- Rejecting repeated outputs. Different inputs may share one output. The violation occurs when one input has different outputs.
- Reading f(x) as multiplication. It means the output of function f at input x.
- Dropping parentheses during substitution. Write f(-3) with every replacement enclosed, especially when powers are present.
- Assuming every formula accepts all real inputs. Check denominators, even roots, logarithms when they appear, and the context.
- Confusing domain and range. Domain values go in; range values come out.
- Using the horizontal line test to decide whether a relation is a function. Use vertical lines for that question. Horizontal lines test whether a function is one-to-one.
- Trusting one representation without checking. A point from the table must satisfy the rule and appear on the graph.
Where Functions Fit in Algebra
Functions turn individual calculations into relationships between changing quantities. The Algebra 1 study guide shows how equations, tables, graphs, systems, exponents, and quadratics build toward this idea.
After the basics, study transformations, inverses, composition, piecewise functions, and specific families. The Algebra 2 guide maps how polynomial, rational, radical, exponential, and logarithmic functions extend the same input-output framework.
A compact review cycle works well:
- State the function condition in words.
- Test a relation in two representations.
- Evaluate one input with careful substitution.
- Find domain and range, including restrictions.
- Check the answer in a table or graph.
How to Practice Functions with Gauth
For practice with coordinate planes, equations as rules, table-to-line reasoning, and basic graph shapes, the Gauth lesson Function Fundamentals can serve as an optional foundation resource. It covers part of this guide's scope and does not replace the relation, notation, domain, range, and function-family work above.
To focus on one weak point, open Gauth Course, choose Create Course, and enter a topic such as “function notation and domain restrictions” plus your level. Complete one activity, then return to a worked example here and verify it without notes.
Sources and Review
OpenStax Precalculus 2e: Functions and Function Notation, accessed October 10, 2026.OpenStax Precalculus 2e: Domain and Range, accessed October 10, 2026.