MODULE 3 – Cheap Research Essays

Fill in the following blanks and submit using the assignment tool in Module 3. You can fill in the blanks and submit as a word document. The second option is to print out the form, write the answers on the printout, scan your answers and submit your work as a PDF file.

I. Relations and Functions
A. A _________________ is a correspondence between two sets A and B such that each element of set A corresponds to one or more elements of set B. Set A is called the _______________ of the relation, and set B is called the ______________ of the relation.
B. A function is a relation such that for each element in the _____________, there is _________________ corresponding element in the _____________.
C. Given the relation defined by the set {(3, 5), (-2, 4), (1,5), (3,8)}:
1. Is the relation a function? _________
2. The domain of the relation is { ______________}.
3. The range of the relation is { _______________}.
D. To determine whether an equation represents a function, we must show that for any value in the domain, there is ________________________ corresponding value in the range.
1. Determine whether the equation 4×2 + y2 = 8 represents a function
a. First solve for y: y2 = _____ – 4×2
y = ±√(8-4x^2 )
b. Note that for any value of x that gives us a positive radicand, there are two possible values for y. For instance, if x = 1, then y = ±√(8-4〖(1)〗^2 ) = + _____. Since the value of 1 in the domain corresponds to both 2 and _____ in the range, the relation is not a function.
II. Function Notation
A. When an equation is explicitly solved for y, we say that “y is a ________________ of x” or that the variable y __________________ on x. Thus, x is the ___________________ variable and y is the ____________________ variable.
B. Given the function f, for any x-value in the domain we denote the y-value or function value as ___________. The symbol f(x) is read as “the ___________ of the function at x” or simply “f ____ x”.
C. Given f(x) = 3x – 5, f(-2) = ______ and f(c + 2) = ______________
III. Domains of Functions
A. The function f(x)=7x^5-3x^4+2x^2-4x+1 is a ________________________ function of degree _______.
1. The domain of every polynomial function is ______________.
B. The quotient of polynomial functions is called a ____________________ function.
1. The domain of a rational function consists of all real numbers except for which the _____________________ equals zero.
C. The domain of the root function g(x)=√(5-x) in interval notation is _____________.
IV. Intercepts
A. A function can have, at most, ________ y-intercept. The y-intercept can be found by evaluating f(_____).
1. Find the y-intercept of f(x)=3x^2+5x-2. ________
B. The x-intercepts, also called real ___________, can be found by finding all real solutions to the equation __________________.
1. Find all the x-intercepts of the function g(x) = x2 – x – 6. _________________ [Hint: g(x) = 0 when x2 – x – 6 = 0.]
V. Finding Intercepts Using a Graphing Calculator
A. (Refer to the Graphing Calculator Tutorials at http://dtc.pima.edu/~dwilliamson/TI/indexti.html. The topics you should view are:
1. Graphing Functions
2. Adjusting the Graphing Window
3. Calc Key – Value (watch the first two minutes and minutes 4:34 – 6:50)
4. Calc Key – Zero (watch the first 6 minutes)
B. Procedure for finding the zeros (x-intercepts) of a function
1. After entering the function using the [Y=] key, select [2nd][Calc].
2. The screen will ask you for a LeftBound.
3. If the cursor is not to the left of the zero (or x-intercept) move it to the left with the left arrow key.
4. Press _______.
5. The screen will then ask you for a RightBound.
6. Move the cursor to the right of the zero (or x-intercept) with the _________ arrow key.
7. Press [Enter].
8. The screen will then display the question “Guess?”
9. Press _________ again.
10. The screen will display the coordinates of the x-intercept.
C. Find the x- and y-intecepts of the function f(x) = x2+3x-5 to the nearest hundredth using a graphing calculator. y-intercept = ___________. x-intercepts are ________ and________.
VI. Relative Maximum and Minimum Values of a Function
A. When a function changes from __________________ to __________________at a point (a, b), then the function is said to have a relative maximum at x = a.
B. When a function changes from __________________ to __________________at a point (a, b), then the function is said to have a relative minimum at x = a.
C. The word “relative” indicates that the function obtains a maximum or minimum value relative to some _________________________. It is not necessarily the maximum (or minimum) value of the function on the entire domain.
VII. Even and Odd Functions
A. A function is even if for every x in the domain, f(x) = _________. Even functions are symmetric about the ______________. For each point (x, y) on the graph, the point __________ is also on the graph.
B. A function is odd if for every x in the domain, f(-x) = _________. Odd functions are symmetric about the ______________. For each point (x, y) on the graph, the point __________ is also on the graph.
C. Determine whether each function is even, odd or neither:
1. f(x)=4x^3-2x _____________
2. g(x)=3x^4+5|x| _____________
VIII. Use the graph of function f to answer the following questions. (estimate the best you can)
1. What is the domain? _______________
2. What is the range? _______________
3. On what intervals is the function increasing? __________________________, decreasing? _____________________________, or constant? ___________________
4. For what value(s) of x does f obtain a relative minimum? ______________. What are the relative minimum values? ____________________
5. For what value(s) of x does f obtain a relative maximum? ______________. What are the relative maximum values? ____________________
6. What are the x-intercepts? ___________
7. What is the y-intercept? __________
8. Is the function even, odd or neither? _________________
9. For what values of x is f(x) > 0? __________________
10. What is f(5)? _________
11. For what value(s) of x is f(x) = -2? ___________
IX. Basic Functions
A. Identify each of the basic functions by name:
1. _______________ function 2. ___________ function

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3. ____________________ 4. ____________ function
function

5. ____________ function 6. ________________
function

7. _______________ function 8. __________________
Function

X. Piecewise-Defined Functions
A. Functions defined by a rule that has more than one piece are called ______________________________ functions.
B. Write the rule that describes the function in the graph.

1. For x < -1, what is the slope of the line? ______
2. If the line were continued, where would it cross the y-axis? ______
3. Therefore the rule for x < -1 is f(x) = _______________
4. Find the point sketched at x = -1. What is f(-1)? ______
5. For x > -1, what is the slope of the line? _______ What is the y-intercept? _______
6. Therefore the rule for x > -1 is f(x) = ___________
7. That gives us the following piecewise-defined function:
f(x)={█(x^2-1″ if ” x<-1@-1″ if ” x=-1@x^2+1″ if ” x>-1)}
XI. Transformations
A. Shifts
1. If c is a positive number, then:
a. the graph of y = f(x) + c is obtained by shifting the graph of y = f(x) _______________________________ a distance of c units,
b. the graph of y = f(x + c) is obtained by shifting the graph of y = f(x) _______________________________ a distance of c units,
c. the graph of y = f(x) – c is obtained by shifting the graph of y = f(x) _______________________________ a distance of c units,
d. the graph of y = f(x – c) is obtained by shifting the graph of y = f(x) _______________________________ a distance of c units.
2. Practice. The graph of:
a. y=√x-6 is the graph of y=√x shifted 6 units _______________.
b. y=(x+3)^3 is the graph of y=x^3 shifted 3 units ______________.
c. y=|x-4|+7 “is the graph of ” y=|x| shifted _______ units horizontally to the right and _____________ units vertically upward.
B. Reflections. For a function defined by y = f(x),
1. The graph of y = -f(x) is a reflection of the graph of f about the ___________.
2. The graph of y = f(-x) is a reflection of the graph of f about the ___________.
C. Stretches and Compressions
1. If a > 1, the graph of y = af(x) is a vertical _______________ of the graph of y = f(x) and is obtained by multiplying each y-coordinate on the graph of f by a factor of a.
2. If 0 < a < 1, the graph of y = af(x) is a vertical _____________________ of the graph of y = f(x) and is obtained by multiplying each y-coordinate on the graph of f by a factor of a.
3. If a > 1, the graph of y = f(ax) is a horizontal _____________________ of the graph of y = f(x) and is obtained by dividing each _____________________ on the graph of f by a factor of a.
4. If 0 < a < 1, the graph of y = f(ax) is a horizontal _____________________ of the graph of y = f(x) and is obtained by dividing each ______________________ on the graph of f by a factor of a.
D. Combinations of Transformations
In order to obtain the graph of y=1/2 √(x+3)-5, the graph of y=√x is shifted ______ units horizontally to the left. This graph is then vertically compressed by a factor of _______. Finally, the graph is shifted ______ units vertically downward.
In order to obtain the graph of y=-|2x|+3,”the graph of ” y=|x| is __________________ compressed by a factor of 2. This graph is then reflected about the ____________. Finally the graph is shifted 3 units ________________________.
XII. Algebra of Functions
A. If f(x) = x + 2 and g(x) = x – 3,
(f+g)(x) = x + 2 + _________.
(f-g)(x) = (x + 2) ____ (x – 3).
(fg)(x) = (x+2)(_______)
(f/g)(x) = (x+2)/(x-3), x ≠ ____.
(f-g)(10) = ______
fg(5) = _______
B. The notation A ∩ B is used to represent the _______________________ of sets A and B.
1. Find [-2, ∞) ∩ (-∞, 5). ________________
C. If f(x)=3x-1 “and ” g(x)=√x, the domain of f+g is _____________ and the domain of f/g is ____________________.
XIII. Composite Functions
If f and g are functions, then the composite function of f and g is defined by
(f ◦ g)(x) = __________ provided ___________ is in the domain of f.
If “f” (“x” )”= x+7 and ” g(x)=8/x , then:
1. (f ◦ g)(x) = 8/x+7
2. (g ◦ f)(x) = _________________
3. (f ◦ g)(2) = ______
4. (g ◦ f)(2) = ______
XIV. One-to-one and Inverse Functions
A. A function is one-to-one if for any two different input values , the corresponding ______________ values must be ___________________.
1. An alternate definition says that if two range values are the same, f(u)= f(v), then the ________________values must be the same; that is, u = _______.
2. If every horizontal line intersects the graph of a function f at most once, then f is ______________________.
3. Is the function f(x) = 5|x| a one-to-one function?
a. Using the first definition, x = 3 and x = -3 are two different input vaules.
b. If f(x) is one-to-one, f(3) must be different from f(_____).
c. f(3) = 5|3| = 5∙3 = 15, f(-3) = 5|-3| = ______
d. since f(3) = f(-3), the output values are not different, thus f is not one-to-one.
4. Use the horizontal line test to decide whether the function graphed is a one-to-one function.
Is this one-to-one? ___________
B. Inverse Functions
1. The domain of f is exactly the same as the _____________ of f -1, and the ____________ of f is the same as the domain of f -1.
2. If the point (a, b) is an ordered pair on the graph of f, then the point (______) must be on the graph of f -1.
3. Cancellation equation for inverse functions: f -1(f(x)) = ______ for all x in the domain of f.
4. The graph of f -1 is a reflection of the graph of f about the line ______________.
5. f -1 exists if an only if the function f is ___________________.