3.2 Functions and Function Notation
*Example 9: Determining Even and Odd Functions Algebraically*
(Brian McLogan) https://www.youtube.com/watch?v=6-B1lkBbmak
*Example 8: Determining Even and Odd functions from a Graph*
(Brian McLogan) https://www.youtube.com/watch?v=QHrEUhUBVrc
*Example 1: Determining whether a Correspondence is a Function*
(Joseph Johnson) https://www.youtube.com/watch?v=YwuRIZ1AIu4
Example 10: Functions and Word Problems*
(Khan Academy) https://www.youtube.com/watch?v=2VEMIoZi1SU https://www.youtube.com/watch?v=PfnLP9ixQY4
*Examples 4,5,6 - Evaluating Functions*
(MathWOES) https://www.youtube.com/watch?v=wvFeAVWHo_Q
*Example 7: Simplifying the Diffrence Quotient*
(The Organic Chemistry Tutor)
*Example 3: Graphing and Vertical Line Test*
(The Organic Chemistry Tutor) https://www.youtube.com/watch?v=DrEXTC6mIO8
*Example 2: Determining Functions with the Vertical Line Test*
(The organic chemistry Tutor) https://www.youtube.com/watch?v=Mxe2lX1htNk
2 potential problems for Domain and Range of a function:
1. Division by zero: the domain of the function y= 2/x-1 is the set of all real numbers other than x=1. When x=1, the denominator is 0, and division by 0 is not defined 2. Even roots of negative numbers: for example, the function y=√x-1 is defined only in the real number system for x≥1 since we exclude the square root of negative numbers. hence, the domain of a function consists of all real numbers x≥1
*Graph of a function f* is defined as the graph of the equation y=f(x), therefore, its possible to use the graph of an equation to test whether to determine a function
We consider all vertical lines, if no vertical line intersects the graph at more than one point, this means that the correspondence used in sketching the graph assigns exactly one y-value for each x-value and therefore determines y as a function of x
*Function, Domain, Image, and Range*
a *function* is a rule that, for each x in set X, assigns exactly one y ina set Y. The element y is called the *image* of x. The set X is called the *domain* of the function, and the set of all images is called the *range* of a function
Wevuse the letter f to designate a function, then we denote the output by responding to x by f(x) which is read "f of x"
frequently, we use the letter y to denote the output corresponding to the input x
The symbol x can also be thought of as holding a ...
place
For each element x in X, there corresponds one and only element y in Y...
that is, the rule assigns exactly one y for a given x. This type of correspondence plays a fundemential role in mathematics called a function
The range of a function is not easily determined as its domain. The range is the set of all y-values that occur in the correspondence, or...
the set of all outputs of a function. Find the range by examining graph of function.
Since we are free to choose the values of x that we drop in the machine, *we call x the INDEPENDENT VARIABLE*;
the value of y that drops out depends upon the choice of x, so y is called the *Dependent Variable*. The dependent variable is a function of the independent variable; that is, the output is a function of the input
*Vertical Line Test*
A graph that represents y as a function of x, if and only if, no vertical line meets the graph at more than one point
*3.2d Symmetry of Functions*
Even Function: graph symmetric about y-axis Odd Function: graph that is symmetric about origin Note: function can never be symmetric about the y-axis because any graph with this symmetry fails the vertical line test
We can think of the rule defined by the equation y=2x+3 as a function machine. Each time we drop a value of x from the domain into the input hopper, exactly one value of y falls out the output chute.
For example, if we drop in x=5, the function machine follows this rule and produces y = 13
A Function: correspondence for each x in X there is one corresponding value of y in Y. The fact that y₁ is an image of both x₁ and x₂. Does not violate the definition of a function.
Not a Function: x₁ has two images assigned to it, namely, y₁ and y₂, thus violating the definition of a function
The equation y=2x+3 assigns y for every value of x. If we let X denote the set of values that the equation assigns to y, we can show the correspondence schematically
The equation can be thought as a rule defining a correspondence from the set X to the set Y
Some vertical lines intersect the graph at two points. Since a correspondence would assign the values to the values y₁ and y₂ to x₁, it does not determine y as a function of x.
Thus, not every equation or correspondence in the variables x and y determines y as a function of x
