PEI Math 521A common mistakes
PEI Math 521A · Understand the traps and how to avoid them.
When solving an equation such as √(2x+5) = x − 1, the candidate squares both sides, keeps the resulting root, and never substitutes back into the original equation to discard the extraneous solution introduced by squaring.
Squaring is treated as a reversible operation, and the candidate has usually been trained on equations where every algebraic step is reversible. Extraneous roots are invisible in the squared equation, so substitution is the only place they surface, and students who are focused on 'getting x' skip the check entirely.
What to do instead
Adopt a fixed routine for any non-reversible step (squaring, multiplying by an expression containing the variable, raising to an even power): after solving, substitute every candidate into the original equation and show the substitution explicitly. Practise with equations engineered to have one genuine and one extraneous root, and record in writing why the extraneous root fails (e.g. the right side is negative for that x).
The candidate applies a function transformation in the wrong order or direction: writing the graph of f(2x − 6) as a shift right 6 then a compression by 2, or graphing f(x + 3) as a shift left by 3 on the wrong side of the axis, or treating −f(x) as a horizontal reflection.
Transformations are memorised as a list of moves without understanding that horizontal changes act on the input before the function is evaluated, so the natural left-to-right reading order of the expression does not match the order of geometric operations. The horizontal direction is also counterintuitive because adding inside the parentheses moves the graph left.
What to do instead
Rewrite every horizontal transformation in the standard form f(k(x − h)) before graphing, factoring the coefficient out of the parentheses. Verify with three test points (a zero, a point where the inside equals 1, and a negative input) rather than trusting memory, and always state the mapping of a specific point such as (0, f(0)) to its image.
When finding the inverse of a function, the candidate swaps x and y, solves for y, and reports the result as the inverse without checking whether the original function is one-to-one on its stated domain, and without restricting the domain when it is not.
The swap-and-solve algorithm is learned as a purely symbolic procedure, and the existence condition (passing the horizontal line test on the given domain) is treated as a separate textbook topic rather than a precondition for the algebra. Quadratic and trigonometric examples then produce 'inverses' that are not functions.
What to do instead
Before any algebra, state the domain and range and confirm one-to-one behaviour on that domain; if the function fails, restrict the domain explicitly and say which branch is chosen (for example x ≥ 0 for x²). After solving, verify with a composition check, f(f⁻¹(x)) = x on the inverse's domain, and confirm that the graph of the inverse is the reflection across y = x.
The candidate solves a trigonometric equation such as sin(2θ) = 1/2 by writing only θ = 15° or a single principal value, ignoring the second solution in the period, the general solution with the period added, and any restriction on the interval given in the problem.
Inverse-trig buttons on a calculator return exactly one value, so the candidate equates 'the answer' with 'the calculator output'. Reference-angle symmetry and periodicity require deliberate reasoning that the calculator conceals, and the doubled angle in sin(2θ) makes the period of the solution set half of what intuition suggests.
What to do instead
Solve in two stages: first find all solutions for the inner argument over a full period using reference angles and quadrant analysis, then divide the entire solution set (including the period term) by the coefficient of θ and filter to the required interval. Always sketch the unit circle or one period of the graph to count how many solutions the interval should contain before listing them.
The candidate applies the Law of Sines to a triangle given two sides and an included angle (SAS) or three sides (SSS), or uses the Law of Cosines when a simple right-triangle ratio applies, and in SSA cases reports only the acute solution and misses the obtuse one.
Both laws look interchangeable in formula sheets, and the decision rule (which information is given) is not tied to a concrete procedure. The ambiguous SSA case is under-practised because most textbook problems are constructed to have a single clean answer.
What to do instead
Use a decision checklist keyed to the given data: ASA/AAS or a known side–opposite-angle pair means Law of Sines; SAS or SSS means Law of Cosines. For SSA, always compute the candidate angle and then its supplement, test both against the angle sum, and state how many triangles exist. Check the final triangle by verifying the angle sum and re-deriving one side with the other law.
When computing a definite integral with substitution, the candidate converts the antiderivative back to the original variable but leaves the original x-limits in place, or changes the limits to u-values and then substitutes x back before evaluating, producing a numerically wrong value with correct-looking work.
The two valid procedures (change limits and stay in u, or keep limits and back-substitute) are each individually simple, but mixing them is easy because the back-substitution step feels like part of 'finishing the problem'. The error is silent: the algebra is all correct, only the evaluation is wrong.
What to do instead
Choose one convention and write it at the top of the solution. If changing limits, compute the new limits immediately after defining u and cross out the old ones on the integral sign. If keeping limits, write 'back-substitute, then evaluate at x = a and x = b' as a literal step. Sanity-check the sign and rough magnitude against an area estimate.
The candidate states a limit exists and equals a finite number when the one-sided limits disagree, when the function is unbounded, or when the expression is an indeterminate form such as 0/0 that was never resolved; alternatively, the candidate differentiates a function to check continuity.
Limit, continuity and differentiability are studied in adjacent chapters and their definitions blur together. Indeterminate forms tempt the candidate to cancel incorrectly or to declare 'undefined' as if it were a limit value, and the distinction between a limit existing and a function being defined at a point is subtle in notation.
What to do instead
For every limit question, first compute the left-hand and right-hand limits separately and compare them explicitly. Classify the form before doing algebra, and only then choose a technique (factor and cancel, rationalise, or compare growth rates). Write the formal criteria as a three-line checklist: limit exists, value equals f(a), and the two-sided limit matches f(a), for continuity; derivative from the limit definition for differentiability.
The candidate interprets a p-value as the probability that the null hypothesis is true, or concludes that a non-significant result proves the null hypothesis, and describes a confidence interval as having a 95% chance of containing the true parameter for this particular sample.
The conditional direction of the p-value is reversed in ordinary speech ('the probability the result is due to chance'), and textbook phrasing is compressed. The correct statement is conditional on the null hypothesis and concerns data, not hypotheses, which is the opposite of what intuition supplies.
What to do instead
Write out the definition in full sentences with the conditional made explicit: the p-value is the probability of data at least as extreme as observed, assuming the null model. Practise rewriting wrong interpretations into correct ones, and phrase confidence intervals as a property of the procedure over repeated sampling, not of the single computed interval.
The candidate computes the number of arrangements of a multiset or a committee with repeated roles by treating identical items as distinguishable (using n! instead of n!/(n₁!n₂!…)), or adds counts where the cases are not mutually exclusive, producing double counting.
The multiplication principle is learned first and applied by default, and the division by factorials for identical objects feels like an arbitrary patch. Overlap between cases is hard to see without a systematic case structure, and small examples are rarely enumerated by hand to check.
What to do instead
Before computing, label each object as distinct or identical and decide whether order matters and whether repetition is allowed, writing the four-way classification explicitly. For case-based counting, define cases by a single discriminating property so they partition the set, then verify with a small analogue (three or four items) enumerated by hand or by listing.
The candidate sets up a modelling problem by grabbing the first formula that mentions the quantities in the prompt, without defining variables, units or constraints, and then reports an answer with no check that it is physically or contextually possible (a negative time, a non-integer number of people, a probability above 1).
Word problems are read for keywords rather than modelled, and the candidate has not been required to justify the choice of function type. Because no variables were defined, the final number has no units attached, so impossible results pass unnoticed.
What to do instead
Use a fixed four-step written protocol: define each variable with its unit and admissible range, state the relationship in words before symbols, derive the equation or function, then interpret the result in context and reject values outside the stated range. Finish by checking the answer against a rough estimate and confirming the units of the final quantity.