Functions and Equations

IB Higher Level Mathematics : Core

2. Functions and Equations

2.1

Concept of function f:x

f(x): domain, range; image (value).

Information on functions can be found at Relations and Functions. This is one part of Interactive Real Analysis, an online, interactive textbook for real analysis or advanced calculus in one real variable from Seton Hall University.

Composite functions f(g(x)); identity function; inverse function f^-1.
Domain restriction.
The graph of a function; its equation y = f(x).

AZ- MATH Software - University of Arizona is a collection of MSDOS and Macintosh software packages written to be used in the teaching of mathematics. Among the packages on offer is composit.zip, which shows how f(g(x)) can be constructed graphically from the graphs of f(x), g(x) and y = x, (below). Recommended.

The interactive DOS package composit.zip, from AZ- MATH Software - University of Arizona, provides a clear demonstration of the composition of two functions. The x-value a is acted upon by the function g. This g(a) value is "bounced" off the line y = x, and is then acted upon by the function f. The resulting value is placed directly above the original x = a value, and represents f(g(x)).

The function and inverse function concepts are explained on the Functions : A Review page of Calculus : An Overview.

The concept of an inverse function is explained at Inverse Functions, from California State University San Bernardino's Reference Notes Page.

Also see One-to-One Functions and Inverse Functions, and How to find the inverse? from Wei-Chi Yang.

A mapping diagram of a function f and its inverse f^-1,
from Inverse Functions. Used with permission.

2.2

Function graphing skills: use of a graphic display calculator to graph a variety of functions.
Appropriate choice of 'window'; use of 'zoom' and 'trace' (or equivalent) to locate points to a given accuracy; use of 'connected' and 'dot' (or equivalent) modes as appropriate.
Solution of f(x) = 0 to a given accuracy.

For a list of links to graphing calculator sites, please refer to the Chinese International School Calculators Page.

2.3

Transformations of graphs: translations; stretches; reflections in the axes.
The graph of f^-1 as the reflection in the line y = x of the graph of f.
Absolute value function | f |.
Graph of 1/f(x) from f(x).

Refer to Syllabus Section 2.1 for inverse graphs.

2.4

The reciprocal function x 1/x, x

0: its graph, its self-inverse nature.

2.5

The quadratic function x ax² + bx + c: its graph.
The form x a(x - h)² + k: vertex (h, k) and y-intercept (0, c).
The form x a(x - p)(x - q): x-intercepts (p, 0) and (q, 0).

For more information on quadratics, please see S.O.S. Mathematics' section on Quadratic Equations on their Algebra page.

2.6

Solution of f(x) = g(x), linear or quadratic.

2.7

Inequalities in one variable, including their graphical representation.
Solution of f(x) g(x), f, g linear or quadratic.

See Solving Absolute Value Inequalities, from the University of Wisconsin Marathon Center's Notes for College Algebra and Geometry.

Also see Inequalities and Graphical Methods for Solving Inequalities, both from the University of Saskatchewan's Exercises in Math Readiness (EMR) site.

2.8

Polynomial functions.
The factor and remainder theorems, with application to the solution of polynomial equations and inequations.

See Intermediate Algebra's chapter on Polynomials.

2.9

The exponential function x a^x, a > 0: its domain and range.
The inverse function x log_a x.
Graphs of y = a^x and y = log_a x.
Solution of a^x = b.

Exponential Functions are fully discussed at the Analysis WebNotes site.

The same site introduces logarithmic functions in a problem : see Homework Assignment 06.

2.10

The functions x e^x, x ln x.
Application to the solution of equations based on problems of growth and decay.

The number e is explained at The Natural Logarithmic Base, and is one of many numbers that make up Favorite Mathematical Constants.