Algebra and Functions
- Syllabus
- 0606–2028–2029
- Section
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- Level
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Recent 5 years
In this section
Topic 1
1. Functions
Objectives in this topic
1.1. Understand function terminology
- Understand the terms: function, domain, range (image set), one-one function, many-one function, inverse function and composition of functions.
- Includes explaining in words why a given relation is a function.
1.2. Find domains and ranges of functions
- Find the domain and range of functions, including inverse and composite functions.
- Restrict the domain of f where needed for f^-1 and/or gf to exist.
- Understand that the domain of gf is a subset of the domain of f, and the range of gf is a subset of the range of g.
1.3. Use function notation
- Recognise and use function notation such as f(x), f: x -> lg x, f^-1(x), fg(x) = f(g(x)) and f^2(x) = f(f(x)).
- The notation f^2(x) will not be used with trigonometric functions.
1.4. Relate y = f(x) and y = |f(x)|
- Understand the relationship between y = f(x) and y = |f(x)| where f(x) may be linear, quadratic, cubic or trigonometric.
- For trigonometric functions, use y = a sin bx + c, y = a cos bx + c or y = a tan bx + c, where a is a positive integer, b is a simple fraction or integer, and c is an integer.
- Fractions for b have denominator 2, 3, 4, 6 or 8 only.
1.5. Explain when inverses do not exist
- Explain in words why a given function does not have an inverse.
1.6. Find inverse functions
- Find the inverse of a one-one function.
- Use correct inverse-function notation.
1.7. Form and use composite functions
- Form and use composite functions.
- Understand that the order of functions is important, so fg may not be the same as gf.
1.8. Sketch functions and inverses
- Use sketch graphs to show the relationship between a function and its inverse.
- Understand that each graph is the reflection of the other in the line y = x.
Topic 2
2. Quadratic functions
Objectives in this topic
2.1. Find maximum and minimum values
- Find the maximum or minimum value of the quadratic function f: x -> ax^2 + bx + c by completing the square or by differentiation.
2.2. Use vertex values to sketch or find range
- Use the maximum or minimum value of f(x) to sketch y = f(x) or determine the range for a given domain.
- Use correct notation to write a domain or range.
2.3. Use discriminant conditions
- Know the conditions for f(x) = 0 to have two real roots, two equal roots or no real roots.
- Relate the discriminant to the roots of the equation.
- Apply the related conditions for a line to intersect a curve, be tangent to a curve or not intersect a curve.
2.4. Solve quadratic equations
- Solve quadratic equations for real roots.
- Use factorisation, the quadratic formula and completing the square.
- The quadratic formula is given in the List of formulas.
- On the calculator paper, correct answers are acceptable without working.
2.5. Solve quadratic inequalities
- Find the solution set for quadratic inequalities graphically or algebraically.
- Write solutions in the correct form, such as -3 < x < 4 or x < 1 or x > 6.
Topic 3
3. Factors of polynomials
Objectives in this topic
3.1. Use remainder and factor theorems
- Know and use the remainder theorem and the factor theorem.
3.2. Find polynomial factors
- Find factors of polynomials.
- For a cubic polynomial, first obtain a product of a linear factor and a quadratic factor, for example by observation or algebraic long division.
3.3. Solve cubic equations
- Solve cubic equations.
Topic 4
4. Equations, inequalities and graphs
Objectives in this topic
4.1. Solve modulus equations
- Solve equations of the type |ax + b| = c, |ax + b| = cx + d, |ax + b| = |cx + d| and |ax^2 + bx + c| = d using algebraic or graphical methods.
- For graphical solutions, draw an accurate graph.
- For algebraic methods, any valid method is acceptable.
4.2. Solve modulus inequalities
- Solve graphically or algebraically inequalities involving k|ax + b| and |ax^2 + bx + c|, including comparisons with constants, linear expressions and other modulus expressions.
- Use k > 0 and c >= 0 or c > 0 where specified.
- For graphical solutions, draw an accurate graph; for algebraic methods, any valid method is acceptable.
4.3. Use substitution for related equations
- Use substitution to form and solve a quadratic equation in order to solve a related equation.
- Identify the appropriate substitution in equations involving powers, logarithms or exponentials.
4.4. Sketch cubic polynomials and moduli
- Sketch graphs of cubic polynomials and their moduli when given as a product of three linear factors.
- Clearly label points of intersection with the coordinate axes.
4.5. Solve cubic inequalities graphically
- Solve graphically cubic inequalities of the form f(x) >= d, f(x) > d, f(x) <= d and f(x) < d where f(x) is a product of three linear factors and d is a constant.
Topic 5
5. Simultaneous equations
Objectives in this topic
5.1. Solve simultaneous equations
- Solve simultaneous equations in two unknowns by elimination or substitution.
- Includes equations such as a line with a quadratic relation, equations involving xy, and fractional equations.
Topic 6
6. Logarithmic and exponential functions
Objectives in this topic
6.1. Use logarithmic and exponential graphs
- Know and use simple properties and graphs of logarithmic and exponential functions, including ln x and e^x.
- Understand that f(x) = e^x and g(x) = ln x are inverse functions.
- Understand the asymptotic nature of logarithmic and exponential graphs and state equations of asymptotes.
- Graphs are limited to y = ke^(nx) + a and y = k ln(ax + b), where n, k, a and b are integers.
- Series expansions are not required.
6.2. Use laws of logarithms
- Know and use the laws of logarithms, including change of base of logarithms.
- Combine logarithmic expressions and convert logarithms between bases, including natural logarithms.
6.3. Solve exponential equations
- Solve equations of the form a^x = b.