Mathematics.

functions

Domain and Range

Algebra I30 minDifficulty3 out of 10

You should know: functions

Overview

The domain of a function is the complete set of input values (typically x-values) for which the function is defined; the range is the complete set of output values (y-values) the function actually produces. Together they describe exactly what a function 'accepts' and 'returns' — essential for knowing where a function's graph exists and what values it can take.

Intuition

Think of a function as a vending machine: the domain is the set of buttons you're allowed to press, and the range is the set of snacks that can actually come out. Some buttons might be broken (excluded from the domain) — like dividing by zero, or taking the square root of a negative number — and some snacks might just never be stocked (excluded from the range), even though the machine works fine otherwise.

Interactive Graph

sqrt(x) — domain restricted to x >= 0

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Formal Definition

Definition

For a function f: domain and range are defined as:

Domain(f)={x:f(x) is defined}\text{Domain}(f) = \{x : f(x) \text{ is defined}\}
Domain
Range(f)={f(x):x∈Domain(f)}\text{Range}(f) = \{f(x) : x \in \text{Domain}(f)\}
Range

Properties

Polynomial domain

Every polynomial function has domain (−∞,∞)\text{Every polynomial function has domain } (-\infty,\infty)

Rational function domain restriction

Exclude x where the denominator=0\text{Exclude } x \text{ where the denominator} = 0

Even-root domain restriction

For g(x)n with n even, require g(x)≥0\text{For } \sqrt[n]{g(x)} \text{ with } n \text{ even, require } g(x)\ge 0

Range of a parabola

y=ax2+bx+c: range is [k,∞) if a>0, (−∞,k] if a<0, where k is the vertex’s y-valuey=ax^2+bx+c: \text{ range is } [k,\infty) \text{ if } a>0, \ (-\infty,k] \text{ if } a<0, \text{ where } k \text{ is the vertex's y-value}

Applications

The domain of a projectile's height function is restricted to the time interval from launch until it returns to the ground.

Worked Examples

  1. The denominator cannot equal zero.

    x−4≠0⇒x≠4x-4\neq 0 \Rightarrow x\neq 4

Answer: Domain: all real numbers except x = 4

Practice Problems

Difficulty 3/10

Find the domain of g(x) = (x+1)/(x²-4).

Difficulty 4/10

Find the range of f(x) = -(x-2)² + 5.

Common Mistakes

Common Mistake

Only checking for zero denominators and forgetting even-index roots also restrict the domain.

Every source of restriction must be checked: denominators that could be zero AND radicands under even roots (square root, fourth root, etc.) that must be non-negative.

Common Mistake

Assuming the range of every function is all real numbers, by analogy with the domain of most polynomials.

The range depends entirely on the function's actual output behavior — e.g. a parabola's range is bounded on one side by its vertex, and √x can never output a negative number.

Summary

  • The domain is the set of valid inputs; the range is the set of resulting outputs.
  • Polynomials have domain (-∞, ∞); rational functions exclude x-values that zero the denominator; even roots require a non-negative radicand.
  • The range of a parabola is bounded by its vertex's y-value, on the side determined by whether it opens up or down.
  • Always check every possible restriction (division by zero, even roots) when finding a domain.

References

  1. WebsiteWikipedia — Domain of a function