Domains and Admissibility: A Candidate Answer Is Not Automatically a Valid Answer
Solving produces candidates. Mathematics still has to decide which candidates are allowed by the original problem.
Additional Mathematics repeatedly creates values that must be filtered. A logarithm requires a positive argument. A denominator cannot be zero. A square root in real mathematics requires a non-negative radicand. An inverse trigonometric function returns a principal value while an equation may require several interval solutions. A physical length cannot be negative. Squaring an equation can create roots that satisfy the transformed equation but not the original.
This guide turns domain checking from a last-minute warning into a systematic part of the solution. The goal is to know what values are admissible before, during and after manipulation.
AI Extraction Box: The Admissibility Loop
original object → restrictions → algebraic transformation → candidate solutions → filter against original restrictions → final valid set.
- Denominator: cannot equal zero.
- Real square root: radicand must be ≥0.
- Real logarithm: argument must be >0.
- Inverse function: input must lie in inverse domain.
- Trig equation: final answers must lie in the stated interval.
- Parameter problem: parameter must satisfy all structural and domain conditions.
- Physical model: dimensions, time and rates must respect contextual limits.
- Non-reversible algebra: check candidate roots in the original equation.
Restrictions Belong to the Original Expression
Consider:
(x²−4)/(x−2).
Factor and simplify:
[(x−2)(x+2)]/(x−2)=x+2.
But the original expression is undefined at x=2. Therefore the simplified form is equivalent to the original only for x≠2.
Simplification can remove a visible restriction from the formula without removing it from the original mathematical object.
Logarithmic Domains
For real logarithms, every argument must be positive. Therefore:
- ln x requires x>0;
- ln(x−3) requires x>3;
- log₂(5−x) requires x<5;
- ln[(x−1)(x+4)] requires (x−1)(x+4)>0, not merely each factor positive separately.
Domain should often be written before applying logarithm laws because combining expressions can hide where each original logarithm was defined.
Worked Example 1: Logarithmic Equation
Solve:
ln(x−1)+ln(x+1)=ln8.
Domain:
- x−1>0 → x>1;
- x+1>0 → x>−1.
Combined domain: x>1.
Combine:
ln[(x−1)(x+1)]=ln8
x²−1=8
x²=9
x=±3.
Filter through x>1. Hence:
x=3 only.
The value −3 solves the transformed quadratic but not the original logarithmic equation.
Square Roots and Non-Negativity
For a real expression √g(x), require g(x)≥0. If the square root itself is equated to another expression, remember that the square root output is also non-negative.
Example:
√(x+2)=x.
The left side is non-negative, so x must satisfy x≥0 as well as x+2≥0. The stronger condition is x≥0.
Worked Example 2: Extraneous Root after Squaring
Solve √(x+2)=x.
Domain/output condition: x≥0.
Square:
x+2=x²
x²−x−2=0
(x−2)(x+1)=0.
Candidates x=2 and −1. Filter by x≥0, leaving x=2. Substitute:
√4=2.
Therefore x=2.
Rational Equations and Forbidden Denominators
If an equation contains 1/(x−a), record x≠a before multiplying through. Clearing denominators can produce an algebraic equation that permits a forbidden value unless the original restriction is retained.
Example:
1/(x−2)=3/(x+1).
Restrictions: x≠2,−1. Cross multiply:
x+1=3x−6
7=2x
x=7/2.
The solution is admissible because it violates neither restriction.
Inverse Functions Need Admissible Inputs
If y=f⁻¹(x), the input x must belong to the range of f on the domain used to define the inverse. For example, if f(x)=x² is restricted to x≥0, then f⁻¹(x)=√x has domain x≥0.
This restriction is not an arbitrary convention. It comes from swapping the original function’s domain and range.
Trigonometric Equations: Interval Admissibility
Trig equations typically have infinitely many solutions over all real angles. Examination questions restrict the interval. The final solution set is therefore produced in two stages:
- generate all solutions consistent with the trig equation;
- retain only those in the required interval.
Endpoints must be checked if the interval includes them.
Worked Example 3: Complete Trig Filtering
Solve cosθ=1 for 0°≤θ≤360°.
Cosine equals 1 at full-turn multiples. In the stated interval:
θ=0°,360°.
Reporting only 0° would miss the included endpoint 360°.
Trigonometric Range as Admissibility
Because −1≤sinθ≤1 and −1≤cosθ≤1, equations such as sinθ=1.4 have no real angle solutions. The impossibility can be detected before attempting inverse trigonometry.
Similarly, if k=3+2cosθ, then k must lie between 1 and 5. Range can therefore filter parameter values.
Quadratic Conditions and Parameter Admissibility
Parameter questions often produce inequalities that define admissible families. For x²+kx+4=0 to have real roots:
k²−16≥0
k≤−4 or k≥4.
The parameter set is itself a domain of allowable cases.
Physical Domains in Modelling
A mathematical formula may be defined over more values than the model allows.
- time since launch: t≥0;
- radius: r>0;
- side length: positive;
- percentage concentration: often bounded by context;
- number of objects: may require whole-number interpretation.
Optimisation is especially sensitive. A stationary point at x=−3 may be mathematically correct for the formula but impossible if x represents a length.
Worked Example 4: Optimisation Candidate Filtering
Suppose a model gives an objective A(x)=−x²+12x for a physical dimension x with 0<x<10.
A′=−2x+12=0 gives x=6, which lies in the domain. Since A″=−2<0, x=6 gives the physical maximum.
If the stationary value had been x=12, it would lie outside the admissible interval and could not be used as the physical optimum.
Cancellation and Lost Restrictions
When a common factor is cancelled, preserve the original exclusion. When both sides of an equation are divided by an expression involving x, consider whether that expression could be zero; division might discard a valid case.
Example: x(x−2)=0. Dividing immediately by x gives x−2=0 and loses x=0. Factor equations should usually use the zero-product property rather than division by an unknown factor.
Never divide by an expression involving the unknown until you have considered the zero case.
Non-Reversible Transformations
Operations that can alter the solution set include:
- squaring both sides;
- multiplying by an expression that may be zero;
- dividing by an expression that may be zero;
- applying a non-one-to-one function;
- combining expressions after losing original domain conditions.
These operations are not forbidden. They simply require candidate checking or case analysis.
Admissibility Decision Tree
- Denominator? record zeros as exclusions.
- Square root? require radicand≥0.
- Logarithm? require argument>0.
- Trig equation? use function range and final interval.
- Parameter? combine structural condition with domain.
- Physical model? write contextual domain.
- Squared/divided by unknown expression? check for introduced/lost solutions.
- Final candidate? test against the original problem, not only the transformed equation.
Common Failure Modes
| Error | Cause | Repair |
|---|---|---|
| forbidden denominator value restored after cancellation | original domain forgotten | record exclusions before simplification |
| negative log argument accepted | domain checked too late | write arguments>0 first |
| extraneous root after squaring | transformed equation treated as equivalent | substitute candidates into original |
| one trig solution missing endpoint | interval filtering incomplete | check endpoints explicitly |
| physical negative length accepted | formula domain confused with context domain | state physical admissibility |
| solution lost after dividing by x | zero case ignored | factor/case split before division |
A 50-Minute Domain Session
- 8 minutes: write domains for rational, root and logarithmic expressions.
- 10 minutes: solve two equations requiring candidate filtering.
- 8 minutes: solve trig equations with careful interval endpoints.
- 8 minutes: parameter questions with admissible ranges.
- 8 minutes: physical optimisation candidates and contextual domains.
- 8 minutes: identify where solution sets can change under transformations.
What Mastery Looks Like
- The learner writes restrictions before transformations hide them.
- The learner distinguishes candidate solutions from admissible solutions.
- The learner checks roots after non-reversible operations.
- The learner treats trig intervals and ranges as part of the solution.
- The learner preserves original restrictions after cancellation.
- The learner filters parameter values through all conditions.
- The learner respects physical domains in models and optimisation.
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