CAPTER 1 Function and Models 1 1 Four

CAPTER 1: Function and Models

1. 1 Four ways to Represent a Function

Function • Wher e D is the domain of the function E is the co-domain of the function

Some Examples of functions: By Rule • • Rows diagram v 1 v 2 v 3 v 4 va v f(a) Graph 4

Graph • Range Domain

Solution :

The way to check the function • • By Definition • By rows diagram • If come out from each element in the domain exactly one row By Graph • Vertical line test: we said the graph is function iff every vertical line intersect only one time with the curve

Example: which one is represent a function and explain why? v 1 v 2 v 1 v 2 1 No Because if x=0 then y = +1 or -1 the element 0 has tow images 2 Yes Because each element in R has only one image 3 Yes Because it come out from each element in the domain exactly one row 4 No Because there is no row come out of “ 2” 5 No Because there is more than one row come out from “ 2”

6) 7) 6 No Because if we draw any vertical line we’ll find tow intersect point 7 Yes Because if you draw any vertical line we’ll find only one intersect point

Piecewise Function The function how's defend in many different domain for example

Some Example for piecewise function:

Some properties of absolute value

Even function • • • Odd function • • •

Even function Odd function Not odd nor even

Probability of even & odd function

Increasing & Decreasing Functions: Increasing • Decreasing •

increasing Decreasing

1. 2 Mathematical Models: Polynomial Rational Algebraic function Root Absolute Exponential function Translated function Logarithmic function Trigonometric function

Algebraic function For example:

Power function: Is any function contain only one term and it is in the formula For example:

Polynomials functions

For example: For example for not polynomial functions:

Some spatial kind of polynomial function Constant function: 1. For example:

2. Unit function: 3. Linear function: For example:

4. Quadratic function: For example:

4. Cubic function: For example:

Rational functions

Ø For example:

Root functions

Ø For example:

• For Example:

Absolute functions

Ø For example:

Classify each function and find the domain:

Solution: Question Kind f(x) Polynomial g(x) h(x) r(x) t(x) q(x) j(x) k(x) l(x) m(x) Domain Polynomial or power Rational or power Constant polynomial Rational Root Algebraic -----------

Example 6 page 14: Find the domain of the following Solution:

Find the following value: Solution:

Find the domain & the Range of the following :

Find the domain of:

Trigonometric Functions It is covered in Appendix D

Exponential Function

Logarithmic function:

1. 3 New Function From Old Function Combination Function

The domain of combination function

Composite Function:

Solutions

Transformation of Functions: