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Decide polynomial inverse existence and produce colliding inputs when it fails · Check your progress
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Produce affine and single-branch quadratic inverses with their complete sets · Check your progress
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Produce polynomial composite rules, domains and ranges in the stated order · Check your progress
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Find complete rational inequality sets with original exclusions · Check your progress
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Find complete linear modulus inequality sets with constant or variable bounds · Check your progress
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Find complete polynomial ranges on stated real input sets · Check your progress
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Decide both polynomial composition orders on their original domains · Check your progress
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Give every asymptote and axis intercept of a rational graph with a linear denominator · Check your progress
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Find the complete range of a rational function with a linear denominator · Check your progress
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Apply an ordered sequence of graph transformations to a rule and a point · Check your progress
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Find the axis intercepts, coordinate restrictions and asymptotes of a standard conic · Check your progress
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Eliminate the parameter of a simple parametric curve and find the coordinates it traces · Check your progress
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Decide polynomial inverse existence and produce colliding inputs when it fails · Check again
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Produce affine and single-branch quadratic inverses with their complete sets · Check again
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Produce polynomial composite rules, domains and ranges in the stated order · Check again
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Find complete rational inequality sets with original exclusions · Check again
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Find complete linear modulus inequality sets with constant or variable bounds · Check again
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Find complete polynomial ranges on stated real input sets · Check again
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Decide both polynomial composition orders on their original domains · Check again
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Give every asymptote and axis intercept of a rational graph with a linear denominator · Check again
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Find the complete range of a rational function with a linear denominator · Check again
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Apply an ordered sequence of graph transformations to a rule and a point · Check again
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Find the axis intercepts, coordinate restrictions and asymptotes of a standard conic · Check again
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Eliminate the parameter of a simple parametric curve and find the coordinates it traces · Check again
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Decide polynomial inverse existence and produce colliding inputs when it fails · Retention check
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Produce affine and single-branch quadratic inverses with their complete sets · Retention check
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Produce polynomial composite rules, domains and ranges in the stated order · Retention check
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Find complete rational inequality sets with original exclusions · Retention check
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Find complete linear modulus inequality sets with constant or variable bounds · Retention check
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Find complete polynomial ranges on stated real input sets · Retention check
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Decide both polynomial composition orders on their original domains · Retention check
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Give every asymptote and axis intercept of a rational graph with a linear denominator · Retention check
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Find the complete range of a rational function with a linear denominator · Retention check
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Apply an ordered sequence of graph transformations to a rule and a point · Retention check
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Find the axis intercepts, coordinate restrictions and asymptotes of a standard conic · Retention check
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Eliminate the parameter of a simple parametric curve and find the coordinates it traces · Retention check
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Practise: Decide polynomial inverse existence and produce colliding inputs when it fails
Practise questions on the skills you are learning.
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Practise: Produce affine and single-branch quadratic inverses with their complete sets
Practise questions on the skills you are learning.
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Practise: Produce polynomial composite rules, domains and ranges in the stated order
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Practise: Find complete rational inequality sets with original exclusions
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Practise: Find complete linear modulus inequality sets with constant or variable bounds
Practise questions on the skills you are learning.
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Practise: Find complete polynomial ranges on stated real input sets
Practise questions on the skills you are learning.
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Practise: Decide both polynomial composition orders on their original domains
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Practise: Give every asymptote and axis intercept of a rational graph with a linear denominator
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Practise: Find the complete range of a rational function with a linear denominator
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Practise: Apply an ordered sequence of graph transformations to a rule and a point
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Practise: Find the axis intercepts, coordinate restrictions and asymptotes of a standard conic
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Practise: Eliminate the parameter of a simple parametric curve and find the coordinates it traces
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Next step
Revision notes
Connect domains and ranges with inverse and composite functions, conics, graph transformations and graphical inequalities.
Key ideas
- Functions and admissible inputs
Distinguish a stated domain from an expression’s natural restrictions and justify the complete range.
- Inverse and composite functions
Choose inverse branches, exchange domain and range, and check the full intermediate set in a composition.
- Conics and rational graph features
Derive coordinate restrictions, distinguish holes from asymptotes, and use equations to verify graph features.
- Graph transformations and reciprocals
Map coordinates and intervals, distinguish absolute-value and reciprocal graphs, and retain parametric branches.
- Rational, absolute and graphical inequalities
Track signs, exclusions and numerical boundaries, then check solutions against the original mathematical model.
From the GCE A-Level H2 Mathematics 2027 syllabus
- Include: function, domain and range; inverse and composite functions; conditions for existence of inverse and composite functions; restricting domain to obtain an inverse function; relationship between a one-to-one function and its inverse graphs. Exclude (fg)⁻¹ = g⁻¹f⁻¹ and restricting domain to obtain a composite function.
- Include: graphing a given function with graphing calculator or graphing software; symmetry, intercepts, turning points and asymptotes for y² = ax, x² = by, x²/a² + y²/b² = 1, x²/a² − y²/b² = 1, y²/b² − x²/a² = 1, y = (ax + b)/(cx + d), y = (ax² + bx + c)/(dx + e); equations of asymptotes and axes of symmetry and restrictions on x and/or y; transformations af(x), f(x) + a, f(x + a), f(ax) and combinations; relating |f(x)|, f(|x|) and 1/f(x) to f(x); simple parametric equations and graphs.
- Include: formulating equations, systems of linear equations or inequalities from a problem situation; solving equations exactly or approximately and systems of linear equations using a graphing calculator or graphing software; inequalities f(x)/g(x) > 0 with linear or quadratic expressions; |x| and |x − a| < b ⇔ a − b < x < a + b, |x − a| > b ⇔ x < a − b or x > a + b; solving inequalities graphically.