Recognition Training
Functions - Mistake Analysis - Set I
Six questions on finding f of g of x and g of f of x and showing they differ in general, finding the inverse of a rational function and stating its domain, the largest domain of a function combining a square root and a fraction, the range of a quadratic by completing the square, describing a sequence of transformations in the correct order, and solving an absolute value equation alongside its related inequality.
Functions - Mistake Analysis - Set II
Six questions on converting a quadratic to vertex form by completing the square and stating the vertex, using the discriminant to find the values of k that give equal roots, constructing a quadratic from two given roots and a point, finding the number and location of intersection points between a line and a parabola including a tangency case, finding the asymptotes and x-intercept of a reciprocal graph, and finding the vertex, range and solutions of an equation for an absolute value function.
Functions - Mistake Analysis - Set III
Six questions on evaluating and solving an exponential function by matching powers of the same base, evaluating a piecewise function at points either side of and exactly on its boundary, solving a composite function equation that produces two solutions, finding the inverse of a function involving a logarithm and stating its domain, proving a function is self-inverse, and modelling the height of a thrown ball with a quadratic to find the maximum height and the time it hits the ground.
The 4 Patterns Behind Every Lost Mark
Reversing the Order of Composite Functions
f of g of x means apply g first, then apply f to the result. g of f of x applies the functions in the opposite order and is generally a different function. Students who compute whichever order is more convenient, or who treat composition as multiplication of the two functions, end up solving a different problem than the one asked.
Confusing an Inverse Function with a Reciprocal
The inverse of a function is found by swapping x and y and solving the equation again for y, not by flipping the function into a fraction of one over itself. The domain of the inverse function is the range of the original function, not a copy of the original domain. Students who take a reciprocal, or who reuse the original domain for the inverse, produce a function that looks plausible but is not the inverse at all.
Missing the Second Solution in Squared or Absolute Value Equations
An equation of the form (something) squared equals a positive number, or an absolute value equals a positive number, always has two solutions, one positive and one negative. Students who take only the positive root, or who solve only one branch of an absolute value equation, find one correct solution and present it as the complete answer, missing a solution the mark scheme requires.
Errors in Completing the Square and Reading Transformations
When completing the square with a leading coefficient other than 1, the constant that comes out of the bracket must be multiplied by that coefficient, not added on directly. When reading transformations from a function such as 2f(x minus 3) plus 1, a shift written as x minus h moves the graph right, not left, and a stretch and a translation applied together are read in the order stretch first, then translate. Getting either of these backwards shifts the vertex, the asymptote, or the image point to the wrong location.
The Full Diagnostic Path
- 75 original exam-style questions across composite functions, inverses, quadratics and graphs
- Full worked solutions with M1/A1/R1 IB mark scheme annotations
- Mistake analysis on every question targeting reversed composition order, inverse versus reciprocal errors, and missing second solutions
- Sections: composite and inverse functions, domain and range, transformations, quadratic functions and the discriminant, exponential and piecewise functions
- IB Examiner commentary per section on where marks are most commonly lost
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