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PREREQUISTE CONCEPTS NEEDED BEFORE YOU START THIS PAGE

  • You have gone through the Summer Study Guide.
  • Undefined numbers or division by 0.
  • Exponents - positive and negative
  • Graphing equations using tables
  • Functions
  • Domain and Range
  • Increasing and decreasing functions
  • Radicals/Roots

 

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THE TOOLKIT FUNCTIONS - ALSO CALLED "PARENT FUNCTIONS"
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In Accelerated Pre-Calculus you will be studying a group of functions called the Toolkit Functions. The Toolkit Functions are functions used to develop more sophisticated functions. If you familiarize yourself with the Toolkit Functions you will have a much easier time taking national exams like the SAT and the Advanced Placement Calculus exam as well as more advanced mathematics courses at a college or a university. Make sure you take time to understand all the information on this page.

If you have gone over your Summer Study Guide you have ALL THE TOOLS necessary for you to understand the following functions!

 

YOUR GOAL IS TO UNDERSTAND AND NOT SIMPLY MEMORIZE!

FUNCTION   CHARACTERISTICS

f(x) = x 

x y (x,y)
-40 -40 (-40, -40)
-20 -20 (-20, -20)
0 0 (0, 0)
20 20 (20, 20)
40 40 (40, 40)
 

x-intercept: x = 0
y-intercept: y = 0
Domain: All reals    
Range: All reals
function is increasing: -∞ < x < +∞

y=x

 

FUNCTION   CHARACTERISTICS

f(x) = x 2 

x y (x,y)
-5 (-5)2 = 25 (-5, 25)
-2 (-2)2 = 4 (-2, 4)
0 (0)2 = 0 (0, 0)
2 (2)2 = 4 (2, 4)
5 (5)2 = 25 (5, 25)
  x-intercept: x = 0
y-intercept: y = 0
domain: All reals
range: y ≥ 0
function is decreasing: -∞ < x < 0
function is increasing: 0 < x < +∞
x squared
TEACHER COMMENT: All functions of the form x n where n is an even positive integer will have the same "look" as the above function and share all the same characteristics.

 

FUNCTION   CHARACTERISTICS

f(x) = x 3

x y (x,y)
-10 (-10)3 = -1000 (-10, -1000)
-6 (-6)3 = -216 (-6, -216)
0 (0)3 = 0 (0, 0)
6 (6)2 = 216 (6, 216)
10 (10)3 = 1000 (10, 1000)
 
x-intercept: x = 0
y-intercept: y = 0
Domain: All reals    
Range: All reals
function is increasing: -∞< x < +∞
x powere 3
TEACHER COMMENT: All functions of the form x n where n is an odd positive integer will have the same "look" as the above function and share all the same characteristics.

 

FUNCTION   CHARACTERISTICS

f(x) = square root

table 1

   x-intercept: x = 0
y-intercept: y = 0
Domain: x ≥ 0    
Range: y ≥ 0
function is increasing: 0 ≤ x < +∞
sqaure root
TEACHER COMMENT: All functions of the form even radicalwhere n is an even positive integer will have the same "look" as the above function and share all the same characteristics.

 

FUNCTION   CHARACTERISTICS

f(x) =cuberoot

graph2

   x-intercept: x = 0
y-intercept: y = 0
Domain: All reals    
Range: All reals
function is increasing: -∞ < x < +∞
cube root graph
TEACHER COMMENT: All functions of the form even radicalwhere n is an odd positive integer will have the same "look" as the above function and share all the same characteristics.

 

FUNCTION   CHARACTERISTICS

f(x )= 1/x

x y (x,y)
-6 1/-6 = -.167 (-6, -.167)
-2 1/-2 = -.5 (-2, -.5)
0 1/0 = undefined -------
2 1/2 = .5 (2, .5)
6 1/6 = .167 (6, .167)

 

  

- Domain: All reals, x ≠ 0   
- Range: All reals,  y ≠ 0
- Function is decreasing: -∞ < x < 0
and 0 < x < +∞
- As the x values approach +∞ the y values approach 0 and are positive
- As the x values approach -∞ the y values approach 0 and are negative.
- As the x values approach 0 from the left the y values approach -∞.
- As the x values approach 0 from the right the y values approach +∞.

onedividex

TEACHER COMMENT: All functions of the form 1/(x n) where n is an odd positive integer will have the same "look" as the above function and share all the same characteristics.

 

FUNCTION   CHARACTERISTICS

f(x ) =one divide vsquared

x y (x,y)
-6 1/(-6 2) = .028 (-6, .028)
-2 1/(-2 2) = .25 (-2, .25)
0 1/(0 2) = undefined -------
2 1/(2 2) = .25 (2, .25)
6 1/(6 2) = (6, .028)

 

  

- Domain: All reals, x ≠ 0   
- Range: y > 0
- Function is increasing: -∞ < x < 0
- Function is decreasing: 0 < x < +∞
- As the x values approach +∞ the y values approach 0 and are positive
- As the x values approach -∞ the y values approach 0 and are positive.
- As the x values approach 0 from the left the y values approach +∞.
- As the x values approach 0 from the right the y values approach +∞.
one div x squares
TEACHER COMMENT: All functions of the form 1/(x n) where n is an even positive integer will have the same "look" as the above function and share all the same characteristics.

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BASIC EXPONENTIAL FUNCTIONS
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The following functions are called the "basic exponential functions." The basic exponential functions are classified into two groups.

GROUP 1
f(x) = b x
where 0 < b < 1

 GROUP 2
f(x) = b x
where b > 1

ex. 1) f(x) = (1/2) x    

ex. 2) f(x) = (3/4) x    

ex. 3) f(x) = (5/7) x    

ex. 4) f(x) = (1/4) x    

Any exponential that is in this group have the following features:
a) no x intercept
b) y intercept is y = 1
c) Domain: All reals
d) Range: y > 0
e) Function is decreasing: -∞ < x < +∞
f) As the x values approach +∞ the y values approach 0 and are positive.

And the exponentials in this group have the following look to their graphs:

exponential base less one

TEACHER COMMENT: As x increases the y values get closer and closer to the y axis but NEVER reach y = 0.

ex. 1) f(x) = (1.4) x    

ex. 2) f(x) = (3) x    

ex. 3) f(x) = (5.7) x    

ex. 4) f(x) = (8) x    

Any exponential that is in this group have the following features:
a) no x intercept
b) y intercept is y = 1
c) Domain: All reals
d) Range: y > 0
e) Function is increasing: -∞ < x < +∞
f) As the x values approach -∞ the y values approach 0 and are positive.

And the exponentials in this group have the following look to their graphs:

exponential base more than one

TEACHER COMMENT: As x decreases the y values get closer and closer to the x axis but NEVER reach y = 0.

 

 

Let me convince you of the above facts by graphing an exponential function from each group.

 

ex. 1) Graph f(x) = (1/2) x is a Group 1 Graph.

x y (x, y)
-2 (1/2) -2 = 1/(1/2)2 = 1/(1/4) = 4 (-2, 4)
-1 (1/2) -1 = 1/(1/2)1 = 1/(1/2) = 2 (-1, 2)
0 (1/2) 0= 1 (0, 1)
1 (1/2) 1 = 1/2 (1, 1/2)
2 (1/2)2 = 1/4 (2, 1/4)

Here is the graph:

exp_example_base_less_one

 

ex. 2) Graph f(x) = 2 x is a Group 2 Graph.

x y (x, y)
-2 2 -2 = 1/(22) = 1/4 (-2, 1/4) or (-2, .25)
-1 2 -1 = 1/(21) = 1/2 (-1, 1/2) or (-1, .5)
0 20 = 1 (0, 1)
1 21 = 2 (1, 2)
2 22 = 4 (2, 4)

Here is the graph:

first exponentail example graph

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WHAT IS e?
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First, "e" is a number - an irrational number. 2. 7182818284590452... is a representation of the number "e." As you may recall an irrational number cannot be expressed as a ratio, i.e. a fraction, of two integers. Another way of thinking about an irrational number is that it is a decimal number that does not terminate or repeat. For example, .333... is rational since it can be expressed as 1/3. The number "e" cannot be expressed like the number .333... can.

So where is the number "e" used? It is used in areas that need to "model" growth and decay. Business and science use this number. Populations seem to increase(growth) and decrease(decay) at certain rates. Radioactive materials like carbon 14 decrease in quantity(decay) over long periods of time. If you take out a loan from a bank, you pay back your loan at a certain rate that grows(growth) a banks profits over the period of your loan. If you make an investment with your money you can get a payback(growth) on your investment. All these various types of growth and decay can represented by formulas that use the number "e."

On your TI there is a button for "e." Push 2nd divide to get the "e" option. Then push the button enter to see the "estimation" of "e." Please remember that the number that you see is NOT "e" but an estimate of "e" since "e" is irrational.

A more sophisticated look at the number "e" coming soon.

ex. 3) Graph f(x) = e x is a Group 2 Graph because "e" is a number greater than 1.

Here is the graph:

x y (x, y)
-3 e -3 = 1/(e3) = .0497870684 (-3,.0497870684)
-2 e -2 = 1/(e2) = .135335283 (-2, .135335283)
-1 e-1 =1/(e1) = .367879441 (-1, .367879441)
0 e 0= 1 (0, 1)
1 e 1= 2.71828183 (1, 2.71828183)
2 e2 = 7.3890561 (1, 7.3890561)
3 e3 = 20.0855369 (1, 20.0855369)

 

e to x

 

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WHERE DOES THE NUMBER e COME FROM?
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TOOLKIT FUNCTIONS VIDEOS

- How To Use The Interactive Toolkit Functions?

 

INTERACTIVE TOOLKIT FUNCTIONS

- fx=x

- xsqaure

- cubed

- cuberoot

- cuberoot

-onedivx

- onedivxsqu

- 2powerx

- one half powerx

- f(x) = ex

- f(x) = log (x)
...This is the common log which is the base 10 log, i.e. log10 (x).

- f(x) = ln (x)
...This is the natural log which is the base e log, i.e. loge (x).

UNIT CIRCLE GRAPHS

- f(θ) = sin(θ)

- f(θ) = cos(θ) 

- f(θ) = tan(θ) = sin(θ) / cos(θ)

- f(θ) = cot(θ) = cos(θ) / sin(θ)

- f(θ) = sec(θ) = 1 / cos(θ)

- f(θ) = csc(θ) = 1 / sin(θ)