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Frances puts $50 in a bank account on Monday
morning every week.
Draw a graph of what Frances's bank account
looks like over time. Put number of weeks on the
horizontal axis, and number of dollars in her
account on the vertical axis.
A)
C)
B)
D)
E)
None of the
Above
?
$
months
Not Correct
$
weeks
Looking at the problem day by day
$
weeks
Solution
C)
$
weeks
Whirlwind Function Tour
Piecewise Functions
Absolute Value
Monomials (AKA Power Functions)
Writing a formula for Frances
Piecewise function
Function definition is given over interval “pieces”
Ex:
Means: “When x is between 0 and 2, use the
formula “2x+1.” When x is between 2 and 5, use
the formula “(x-3)2”
Consider the piecewise function below:
Find f(3).
A) 7
B) -2
C) -3
D) Both (a) and (b)
E) None of the above
Solution
Find f(3)
When x is less than or equal to 5, we use the
formula “x-5”
Three is less than or equal to five.
3-5=-2
B) -2
Graphing
“When x is between 0 and 2, use the formula “2x+1.”
When x is between 2 and 5, use the formula “(x-3)2”
Special Piecewise Functions
floor(x) (pronounced “floor of x”)
Writes functions like Frances' easily
Used a lot in computer programming
Not covered in this class
Ask me if you're curious
abs(x) (pronounced “the absolute value of x”)
|x|
Also written abs(x)
Pronounced “The absolute value of x”
Definition
Monomials
Monomial (Power Function)
Any function of the form: f(x)=axb
(These are not exponentials: f(x)=abx)
Monomials have five sub families:
Even power
Odd power
1/n Power
Negative Power
Real Power
Making a Table
If you want to know how a monomial behaves,
you can ALWAYS make a table.
f(x)=2x3
x
f(x)
-2
-16
-1
-2
0
0
1
2
2
16
Even Power Monomials
Power of 0,2,4,6,8,etc.
y never changes sign
f(x)=3
f(x)=3x0
g(x)=x2
h(x)=-0.5x4
Odd Power Monomials
Power of 1,3,5,7,etc.
y changes sign at x=0
f(x)=2x
f(x)=x3
f(x)=-2x5
1/n Power Monomials
(Root functions)
Powers are fractions: 1/2, 1/3, 1/4, etc.
Powers of 1/n are called the n-th root.
These are inverses of even or odd power
monomials.
½ Power
Domain: x≥0
Range: y≥0
1/3 Power
Domain: all real numbers
Range: all real numbers
Negative Power
Power of -1, -2, -3.572, etc
Have a “hole” in the domain at x=0.
Can't divide by zero!
Negative Power
Domain: x≠0
Range: y>0
Domain: x≠0
Range: y≠0
Notice that the Even/Odd power rules about y changing sign still work
Real Power
Power of any real number: 2/3, 8/5, 3.14159265,
etc.
Uses combinations of the behaviors above
Can be tricky
Make a table!
The average cost, in dollars, of producing the x photo
albums is given by:
Here, x is the number of photo albums. What is the
average cost of producing 500 photo albums?
A) $6.75
B) $5.50
C) $4.50
D) $3.50
E) None of the above
Solution
A(x)=4.5+1000/x
A(500)
4.5+1000/500=4.5+2=6.5
$6.50
E) None of the Above