Functions Introduction and Properties

Mathematics and Statistics Department

• Nova Lisbon-Cairo Campus

Functions Preliminaries and Properties

• Presented by Dr. Omar M. Sallah
• Assistant Professor of Engineering Mathematics
• Department of Mathematics, Nova University of Lisbon
• Contact email addresses provided

Course Notes and References

• Accessible on Moodle

Introduction to Functions

• Functions are crucial in various fields (science, engineering, etc.)
• A function is a relationship between at least two variables
• Independent variable (maps to a number)
• Dependent variable (maps to another unique number)
• Simple valued function represented as y = f(x)
• x is the independent variable, y is the dependent variable
• The relationship between variables defined by the shape of f(x)
• Example: f(x) = x + 1, f(3) = 4, f(8) = 9

Functions and Relations

• A function is a specific type of relation
• Every independent variable has only one value for the dependent variable
• Vertical line test: if a vertical line intersects the graph in more than one point, it's a relation, not a function
• Examples of functions and relations, along with their visual representations (graphs) provided for illustration.

Domain of a Function

• Domain is all possible input values (x-values) for which the function produces a valid output.

Range of a Function

• Range is all possible output values (y-values) that result from the function.

Injective Functions (One-to-One Functions)

• For every output, there's only one input
• Horizontal line test: if a horizontal line intersects the graph more than once, it's not one-to-one.
• Examples of one-to-one and non-one-to-one functions given

Famous Functions and Their Domain and Range

• Linear functions
• Quadratic functions
• Rational functions
• Exponential functions
• Circles (as a special case)
• Detailed descriptions, formulas, and visual examples provided

Function Inverse

• Finding the inverse of a function is a crucial process in mathematics.
• Injective functions have inverses.
• The inverse is found by swapping roles of x and y, to get x in terms of y
• In examples, functions drawn to show symmetric relationships along the y=x axis.

Composite Functions

• Functions can be nested or composed within each other
• The order of functions in a composite function matters (f(g(x)) may be different from g(f(x)))
• Steps to find the composite function of two functions
• Example of composite functions are provided with detailed illustrations.

Further Remarks on Domain and Range

• Important considerations (vertical asymptotes, denominators, and restrictions)
• Special functions(like logarithmic functions or absolute value functions) with their respective characteristics

Examples of Functions and their Properties

• Detailed calculation examples provided for the various functions, which illustrate how to compute using certain values and find the domain and ranges.
• The graphs illustrate the relationship between the functions noted..