Functions and equations

• Concept of function f : x--> f (x).
• Domain, range; image (value).
• Composite functions.
• Identity function. Inverse function f^-1

Not required:

• domain restriction.

• 
• On examination papers, students will only be asked to find the inverse of a one-to-one function.

• Int: The development of functions, Rene Descartes (France), Gottfried Wilhelm Leibniz (Germany) and Leonhard Euler (Switzerland).
• TOK: Is zero the same as “nothing”?
• TOK: Is mathematics a formal language?

• The graph of a function; its equation y = f (x).
• Function graphing skills.
• Investigation of key features of graphs, such as maximum and minimum values, intercepts,horizontal and vertical asymptotes, symmetry,and consideration of domain and range.
• Use of technology to graph a variety of functions, including ones not specifically mentioned.
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• Note the difference in the command terms “draw” and “sketch”.
• An analytic approach is also expected for simple functions, including all those listed under topic 2.
• Link to 6.3, local maximum and minimum points.

• Appl: Chemistry 11.3.1 (sketching and interpreting graphs); geographic skills.
• TOK: How accurate is a visual representation of a mathematical concept? (Limits of graphs in delivering information about functions and phenomena in general, relevance of modes of representation.)

• Transformations of graphs.
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• Composite transformations.

• Technology should be used to investigate these transformations.
• Translation by the vector  denotes horizontal shift of 3 units to the right, and vertical shift of 2 down.
• Example:y = x² used to obtain y = 3x² + 2 by a stretch of scale factor 3 in the y-direction followed by a translation of .

• Candidates are expected to be able to change from one form to another.
• Links to 2.3, transformations; 2.7, quadratic equations.

• Appl: Chemistry 17.2 (equilibrium law).
• Appl: Physics 2.1 (kinematics).
• Appl: Physics 4.2 (simple harmonic motion).
• Appl: Physics 9.1 (HL only) (projectile motion).

• The reciprocal function  its graph and self-inverse nature.
• The rational function  and its graph.
• Vertical and horizontal asymptotes.

Diagrams should include all asymptotes and intercepts.

• Solving equations, both graphically and analytically.
• Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach.
• Solving ax² + bx + c = 0 , a ≠ 0 ..
• The quadratic formula.
• The discriminantand Δ = b² − 4ac the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots.
• Solving exponential equations.

• Solutions may be referred to as roots of equations or zeros of functions.
• Links to 2.2, function graphing skills; and 2.3– 2.6, equations involving specific functions.
• Examples:e^x = sin x, x4 + 5x − 6 = 0  .
• Example: Find k given that the equation has two equal real roots.
• Examples:
• Link to 1.2, exponents and logarithms.

• Appl: Compound interest, growth and decay; projectile motion; braking distance; electrical circuits.
• Appl: Physics 7.2.7–7.2.9, 13.2.5, 13.2.6,13.2.8 Physics,(radioactive decay and half-life)