Lecture Notes For October 29, 1998
I. Return Quiz 2
II. Homework
III. Review For Midterm II
A) 3x3 systems
Example:
x + 2y - z = 2
2x + 2y + 2z = 12
x - y + 2z = 5
Step 1: Choose the most convenient variable to eliminate
In our example any seem convenient
Step 2: Use any two of the equation to eliminate the variable. Then use a different pair to eliminate the same variable. The result is a 2 by 2 system
Multiply equation 1 by two and subtract from equation 2:
2x + 2y + 2z = 12
2x + 4y - 2z = 4
-------------------
-2y + 4z = 8
and
Multiply equation 3 by 2 and subtract from equation 2
2x + 2y + 2z = 12
2x -2y + 4z = 10
------------------
4y - 2z = 2
Step 3: Use elimination to solve the system of the two equations that you found.
2y + 4z = 8
4y - 2z = 2
We multiply the second equation by 2 and add it to the first equation:
-2y + 4z = 8
8y - 4z = 4
----------------
6y = 12
y = 2
Resubstituting:
-2(2) + 4z = 8 so that 4z = 12, z = 3
Step 4: Substitute the two values into any of the three equation the get the third value.
x + 2(2) - (3) = 2 so x = 1
Step 5: substitute all three values into the three equations to check your work.
Step 6: Reread the question and answer it.
B) Nonlinear Systems Using Elimination or Substitution
C) Relations and Functions, Domain and Range and the Vertical Line Test
D) Functional Notation, Composition of Function and Functional Arithmetic
E) Four Important Functions: Linear Functions, Quadratic Functions, Square Root Functions and, Absolute Value Functions
F) Inverse Functions:
For functions, we follow the steps below to find the inverse:
Step 1)
Switch the x and y.
Step 2) Solve for y
Step 3) Write in inverse notation
1-1 functions and the horizontal line test
IV. Key for Practice Midterm II.
Problem 1:
(2,1,3)
Problem 2:
(1,sqrt 8), (-1,sqrt 8), (1,-sqrt 8), (-1,-sqrt 8)
Problem 3
Let the domain be the set of mothers andthe range be the set of children. Then let the rule assign to each mother her children.
Problem 4
{x|x not 1, x not -14}
Problem 5
Functions: A and B
1-1 functions: B
Problem 6
Problem 7
A. Not a function not 1-1
B. A 1-1 Function
C. A funciton but not 1-1
Problem 8
Problem 9
A. 4x2 - 2x
B. 2x + h + 1
C. 3
D. -3
E. 3x2 -3x + 3