This document provides an overview of functions and related concepts in precalculus including: - Definitions of absolute value, even and odd functions, and representing functions using tables, graphs, and formulas - Domains and ranges of functions and how algebraic operations can affect domains - Composition of functions defined as f(g(x)) and the domain being values where g(x) is in the domain of f - Examples are provided to illustrate key concepts like evaluating functions, combining functions, and function composition.

1. Jamhuriya university for science and technology
• Course: Calculus I
• Class CE201
• Chapter 1 Precalculus
• Topic : functions
• Faculty Of Engineering
• Lecturer: Eng Abdirahman Farah Ali

2. properties of absolute value
If a and b are real numbers, then
Recall that the absolute value or magnitude of a real number x is defined by
The effect of taking absolute value of a number is to strip away the minus sign if the number is negative and to leave the
number unchanged if it is nonnegative.
Examples:
THE ABSOLUTE VALUE FUNCTION

3. DOMAIN AND RANGE

4. continue

5. definition
If a real-valued function of a real variable is defined by a
formula, and if no domain is stated explicitly, then it is to be
understood that the domain consists of all real numbers for
which the formula yields a real value. This is called the
natural domain of the function

6. Example

7. continue

8. THE EFFECT OF ALGEBRAIC OPERATIONS ON THE DOMAIN
Algebraic expressions are frequently simplified by canceling common factors in
the numerator and denominator. However, care must be exercised when
simplifying formulas for functions in this way, since this process can alter the
domain.
f(x) = x + 2, x= 2

10. • Infinity has no end
unlimited, endless, without bound. The common symbol for infinity, ∞,
was invented by the English mathematician John Wallis in 1655.
The issue of infinitely small numbers led to the discovery of calculus in
the late 1600s by the English mathematician Isaac Newton and the
German mathematician Gottfried Wilhelm Leibniz. Newton introduced
his own theory of infinitely small numbers, or infinitesimals, to justify
the calculation of derivatives, or slopes. In order to find the slope (that
is, the chansge in y over the change in x) for a line touching a curve at a
given point (x, y), he found it useful to look at the ratio
between dy and dx, where dy is an infinitesimal change in y produced by
moving an infinitesimal amount dx from x
Infinity

11. • Infinity represents something that is boundless or endless, or
else something that is larger than any real or natural number.
• It is often denoted by the infinity symbol shown here.

12. Even and Odd Functions
Definition:
• If f (x) = f (−x) for all x in the domain of f , then f is an even function.
• If f (−x) = − f (x) for all x in the domain of f , then f is an odd function.
Example: Determine whether each of the following functions is even, odd, or neither
• la. f (x) = −5x4 + 7x2 − 2
• b. f (x) = 2x5 − 4x + 5
• c. f (x) =
3x
x2 + 1

13. Try this:
Determine whether
f (x) = 4x3 − 5x is even, odd, or neither.

14. NEW FUNCTIONS FROM OLD
• ARITHMETIC OPERATIONS ON FUNCTIONS
• Two functions, f and g, can be added, subtracted, multiplied, and divided in a natural way to form new
functions f + g, f − g, f*g, and f /g.
For the functions f + g, f − g, and fg we define the domain to be the intersection
of the domains of f and g, and for the function f /g we define the domain to be the
intersection of the domains of f and g but with the points where g(x) = 0 excluded (to
avoid division by zero).

15. Example
• Let
• Find the domains and formulas for the functions f + g, f − g, fg, f /g, and
7f .
• Solution. First, we will find the formulas and then the domains. The formulas are

16. Domain

17. Representing Functions
• Representing Functions
• Typically, a function is represented using one or more of the following
tools:
• • A table
• • A graph
• • A formula

18. COMPOSITION OF FUNCTIONS
• We now consider an operation on functions, called composition, which has no direct analog in ordinary
arithmetic. Informally stated, the operation of composition is performed by substituting some function for the
independent variable of another function. For example,
suppose that.

19. definition
• Given functions f and g, the composition of f with g, denoted by f ◦g , is the function defined by
(f ◦g)(x) = f(g(x)) The domain of f ◦g is defined to consist of all x in the domain of g for which g(x)
is in the domain of f .

21. TRY THIS

22. Assignment #1
1- Given f (x) = − x2 + 6x −11 find each of the following.
• (a) f (2)
• (b) f (−10)
• (c) f (t )
• (d) f (t − 3)
• (e) f (x − 3)
• (f) f (4x −1)
2- Given the functions f (x) = 2x − 3 and g(x) = x2 − 1, find each of the
following functions and state its domain.
• a. ( f + g)(x)
• b. ( f − g)(x)
• c. ( f · g)(x)
• d. f/g(x)