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Chapter 23. Algebra of Functions. What will be taught ?. 1. Some simple notations. 2 . Simple graphs you must known. 3. Relation and function. 4 . Domain and range. 5 . Composite function. 6 . Inverse function. 什么意思？. Some basic graphs you must bear in mind !. Relation and Function.
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Chapter 23 Algebra of Functions FYHS-Kulai, Chtan 2011
What will be taught ? 1. Some simple notations 2. Simple graphs you must known 3. Relation and function 4. Domain and range 5. Composite function 6. Inverse function FYHS-Kulai, Chtan 2011
什么意思？ FYHS-Kulai, Chtan 2011
Some basic graphs you must bear in mind ! FYHS-Kulai, Chtan 2011
Relation and Function FYHS-Kulai, Chtan 2011
Ordered pair notation rule domain FYHS-Kulai, Chtan 2011
Mapping notation f Y X x f(x) Domain Range FYHS-Kulai, Chtan 2011
1 a 2 b c 1-1 function FYHS-Kulai, Chtan 2011
1 a 2 b 3 onto function FYHS-Kulai, Chtan 2011
1 a 2 b 3 c 1-1 and onto function FYHS-Kulai, Chtan 2011
1 a 2 b Many to 1 function FYHS-Kulai, Chtan 2011
1 a 2 b Not a function FYHS-Kulai, Chtan 2011
1 a 2 b Not a function FYHS-Kulai, Chtan 2011
Domain and Range of a Function FYHS-Kulai, Chtan 2011
The domain of a function is the complete set of possible values of the independent variable in the function. In plain English, this definition means: The domain of a function is the set of all possible x values which will make the function "work" and will output real y-values. When finding the domain, remember: •The denominator (bottom) of a fraction cannot be zero •The values under a square root sign must be positive FYHS-Kulai, Chtan 2011
e.g.1 The function y = √(x + 4) has the following graph. Note! FYHS-Kulai, Chtan 2011
The range of a function is the complete set of all possible resulting values of the dependent variable of a function, after we have substituted the values in the domain. In plain English, the definition means: The range of a function is the possible y values of a function that result when we substitute all the possible x-values into the function. When finding the range, remember: •Substitute different x-values into the expression for y to see what is happening •Make sure you look for minimum and maximum values of y •Draw a sketch! FYHS-Kulai, Chtan 2011
e.g.2 Let's return to the example above, y = √(x + 4). We notice that there are only positive y-values. There is no value of x that we can find such that we will get a negative value of y. We say that the range for this function is y ≥ 0 FYHS-Kulai, Chtan 2011
e.g.3 The curve of y = sin x shows the range to be betweeen −1 and 1. FYHS-Kulai, Chtan 2011
e.g.4 Find the domain and range of y=tan x. FYHS-Kulai, Chtan 2011
Minor test 2 Sketch the graph . State its range. 2. Sketch the graph . What is the domain and range of this curve. FYHS-Kulai, Chtan 2011
Implied domain or maximal domain Let see the following 3 functions : 𝒈 h FYHS-Kulai, Chtan 2011
g and h are called restrictions of f since their domain are subsets of the domain of f . FYHS-Kulai, Chtan 2011
h(x) g(x) f(x) 4 1 x x x 2 1 0 0 0 Half curve Full curve Three points FYHS-Kulai, Chtan 2011
Codomain What can go into a function is called the Domain. What may possibly come out of a function is called the Codomain. What actually comes out of a function is called the Range. FYHS-Kulai, Chtan 2011
Let see a simple example : x 2x+1 1 B A 2 1 3 Range or image 4 2 5 3 6 7 4 8 9 Domain 10 Codomain FYHS-Kulai, Chtan 2011
Odd and Even functions Odd function e.g. Even function e.g. FYHS-Kulai, Chtan 2011
The graphs of even functions are symmetrical about y-axis. Even functions are many to one functions. Most functions are neither even nor odd. FYHS-Kulai, Chtan 2011
The elementary operations of arithmetic FYHS-Kulai, Chtan 2011
The domain of is : FYHS-Kulai, Chtan 2011
Monotone increasing and monotone decreasing FYHS-Kulai, Chtan 2011
Composite functions FYHS-Kulai, Chtan 2011
f g X Y Z a 1 # b 2 % c 3 * d 4 This is gof, the composition of f and g. For example gof(c)=# FYHS-Kulai, Chtan 2011
The fundamental condition for the existence of composite functions : FYHS-Kulai, Chtan 2011
e.g.5 Given , FYHS-Kulai, Chtan 2011
e.g.6 Given that . Find FYHS-Kulai, Chtan 2011
e.g.7 If function f and g are defined as follows : and FYHS-Kulai, Chtan 2011
Inverse functions FYHS-Kulai, Chtan 2011
f a 1 b 2 3 c a 1 b 2 c 3 FYHS-Kulai, Chtan 2011
Do all one-to-one functions have inverses? Yes. If f is one-to-one, then the flipping operation results in a graph that passes the vertical line test, and so is the graph of a function. A little thought will convince you that this function undoes what the function f did-in other words, that the new function is the inverse of f. FYHS-Kulai, Chtan 2011
Graphing the Inverse of a One-to-One Function If f is a 1-1 function, then it has an inverse f-1. If f is not 1-1, then it does not have an inverse. The domain of f-1 is the same as the range of f; the range of f-1 is the same as the domain of f. To obtain the graph of f-1 from that of f, flip the graph of f about the line y = x, so that the x-axis is superimposed on the old y-axis and vice-versa. The graph obtained is then the graph of f-1. Points on the graph of f-1 are obtained from corresponding points on the graph of f by switching the x- and y-coordinates. FYHS-Kulai, Chtan 2011
Exponential functions and Logarithmic functions y y (0,1) (0,1) x x 0 0 FYHS-Kulai, Chtan 2011
y (0,1) x 0 FYHS-Kulai, Chtan 2011
