About this topic
Before you begin
Connect domains and ranges with inverse and composite functions, conics, graph transformations and graphical inequalities.
Learning goals
- Find complete polynomial ranges on stated real input sets
- Interpret and draw graphs of functions
- Determine function domains, ranges and admissible inputs
- Decide polynomial inverse existence and produce colliding inputs when it fails
- Produce affine and single-branch quadratic inverses with their complete sets
- Produce polynomial composite rules, domains and ranges in the stated order
- Decide both polynomial composition orders on their original domains
- Compose and invert functions with existence conditions
- Give every asymptote and axis intercept of a rational graph with a linear denominator
- Find the complete range of a rational function with a linear denominator
- Find the axis intercepts, coordinate restrictions and asymptotes of a standard conic
- Apply an ordered sequence of graph transformations to a rule and a point
- Eliminate the parameter of a simple parametric curve and find the coordinates it traces
- Transform graphs and relate representations
- Find complete rational inequality sets with original exclusions
- Find complete linear modulus inequality sets with constant or variable bounds
- Formulate, use and interpret a mathematical model
- Solve inequalities and represent their solution sets
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.
Choose a lesson
Work through them in order.
- Not startedFunctions and admissible inputsNextDistinguish a stated domain from an expression’s natural restrictions and justify the complete range.
- Not startedInverse and composite functionsNextChoose inverse branches, exchange domain and range, and check the full intermediate set in a composition.
- Not startedConics and rational graph featuresNextDerive coordinate restrictions, distinguish holes from asymptotes, and use equations to verify graph features.
- Not startedGraph transformations and reciprocalsNextMap coordinates and intervals, distinguish absolute-value and reciprocal graphs, and retain parametric branches.
- Not startedRational, absolute and graphical inequalitiesNextTrack signs, exclusions and numerical boundaries, then check solutions against the original mathematical model.