Functions and graphs
Build and connect the representations and methods in functions and graphs.
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Before you begin
Build and connect the representations and methods in functions and graphs.
Learning goals
- Determine function domains, ranges and admissible inputs
- Interpret and draw graphs of functions
- Compose and invert functions with existence conditions
- Transform graphs and relate representations
- Solve inequalities and represent their solution sets
- Formulate, use and interpret a mathematical model
Syllabus statements covered
- 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.
- 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.
Lessons
Work through them in order.
About lesson progress
Lesson marks record what you have studied. Checks show understanding separately. Guest marks come from this device.
- Functions and admissible inputsDistinguish a stated domain from an expression’s natural restrictions and justify the complete range.
- Inverse and composite functionsChoose inverse branches, exchange domain and range, and check the full intermediate set in a composition.
- Conics and rational graph featuresDerive coordinate restrictions, distinguish holes from asymptotes, and use equations to verify graph features.
- Graph transformations and reciprocalsMap coordinates and intervals, distinguish absolute-value and reciprocal graphs, and retain parametric branches.
- Rational, absolute and graphical inequalitiesTrack signs, exclusions and numerical boundaries, then check solutions against the original mathematical model.