Nonlinear Systems Relations and Functions

I.  Hand Back Midterm I

II.  Homework

III.  Nonlinear Systems

To solve a system of two equations and two unknowns when the equations are not linear, we use the methods of substitution or elimination and hope that the resulting equation becomes a hidden quadratic or other solvable equation.   

Example:  

x² - y = 0

x² - 2x + y = 6

We use substitution: solving for y in the first equation we get:

y = x²

Putting it into the second equation, we have:

x² - 2x + x² = 6

so that

2x² -2x - 6 = 0

or

x² - x - 3 = 0

The quadratic formula gives us:

(1 +- sqrt(1 + 12))/2

so x = 2.3 or x = -1.3

since

y= x²

we get y = 5.29 or y = 1.69

We arrive at the two points:

(2.3,5.29) and (-1.3,1.69)

Example 2:

x² + y² = 29

x² - y² = 3

We use elimination:  adding the two equations, we have

2x² = 32

x² = 16

x = 4 or x = -4

Putting x = 4 into equation 2, we have:

16 - y² = 3

-y² = -13

y = sqrt(13) or y = -sqrt 13

 Putting x = -4 into equation 2, we have:

16 - y² = 3

-y² = -13

y = sqrt(13) or y = -sqrt 13

Therefore we obtain the four points:

(4,sqrt(13)), (4,-sqrt(13)), (-4,sqrt(13)), (-4,-sqrt(13))

The class will try the following examples:

1)

 x² - y² = 4

2x² + y² = 16

2)

y - 2x² = 0

x² +6x - y = 6

3)  

 x² - y² = 21

 x²  + xy - y² = 31

IV.  Relations

A relation is a rule that takes an input from a set (called the range) and gives one or more outputs of another set (called the range).

Examples:

(0,0), (4,4), (0,3), (2,1)

y = 2x

Arrow notation will be given.

We will show how the graph of a circle is a relation.

V.  Functions

A function is a relation such that for every input there is exactly one output.

Example:

(2,1), (3,4), (5,7)

Arrow diagrams will also be given

y = x²

VI.  Determining the Domain of a Function

Rule 1)  The domain of a polynomial is the set of all real numbers.

Example:  for f(x) = 3x² + 2x - 1

the domain is the set of all real numbers.

( more examples will be given in class) 

Rule 2)  The domain of a rational function (poly)/(poly) is the set of all real numbers except where the denominator is 0

Example:  For f(x) = (x - 1)/(x + 1) the domain is all real numbers except where x = -1.

( more examples will be given in class)

Rule 3)  The domain of a square function is all real numbers that make the inside of the square root positive.

Example:  Find the domain of sqrt(x² - x - 6):

Solution:  We set up  the inequality x² - x - 6 > 0 and use our steps of quadratic inequalities to solve. Factoring we get

(x - 3)(x + 2) > 0

Putting -2 and 3 on a number line, gives three regions.  The table shows

	x+2	x - 3	Total
Left (-3)	-	-	+
Middle(0)	+	-	-
Right(4)	+	+	+

Hence the solution is (-infinity,-2] U [3,infinity)

VII  The Vertical Line Test

If every vertical line passes through the graph at most once then the graph is the graph of a function.  Examples will be given in class.