4\5
14]\]1.1 Four ways to represent a function.
1. Visually -> a graph
2. Verbally -> words EX: Two times a numb plus four
3. Numericly-> table of values EX:
x
-2
-1
0
f(x)
0
2
4
4. Algebraically -> with a Formula EX: 2x+4
​
-​
A function f is a rule that assigns each element x in set d to exactly one element which is called
f(x), in set E
-​ D is the Domain of the Function
-​ The range of f is a subset of E
-​ Contains all possible values
-​ f(x) is the dependent variable
-​ X is the independent Variable
The most common method of visualizing a function is with a graph, if F is a function with domain D its
graph is a set of ordered pairs
​
{(x,f(x)| x c D}
Ex: Ex(6) Domain{x| [0,7]}
[PASTE EXAMPLE HERE]​
​
​
Vertical Line Test: A curve in the xy-plane in the graph of a function if and only if no vertical line
intersect with the curve more than once
Piecewise-Defined Functions
Are defined by different formulas in different parts of the domain
A Function is defined by
{1-x of x < -1
{x^2 if x >-1
SYMMETRY
Even Function satisfy f(x) =f(-x) for every x in the domain about the Y axis
Odd functions satisfy f(-x)=-f(x)
INCREASING AND DECREASING FUNCTIONS
Visual
A function is said to be increasing on interval I IF for any x1<x2
A function is said to be decreasing in intervol I if fro any x1<x2, f(x1),f(x2)
1.2 MATHEMATICAL MODELS, A Catalog of Essential Functions
Mathematical Models are descriptions of a real world occurrence
-​ Population size over time
-​ Speed of a falling object
-​ Cost of production
Liner Models
-​ Function whose graph is a line using slope-intercepts from to get y=mx+b where m is the slope
and b is the Y intercept
POWER FUNCTIONS of the form f(x)=x^a where a is a constant
-​ These behave differently depending on a
-​ -a=n is a positive integer
-​ If n is even is “similar” to a parabola
-​ If n is odd, x^n is “similar” to a cubic graph
-​ As n increases, the graph becomes flatter towards zero and sleeper for |x|>1
-​ a= 1/n, n is positive (0<a<1), f(x)= x^1/n =\n/x|
RATIONAL FUNCTIONS ratio of polynomials f(x), where
2
𝑔(𝑥) =
𝑥 +2𝑥+2
𝑥−3
+ (𝑥 − 3) 𝑥 + 3
ALGEBRAIC FUNCTIONS constructed with algebraic operations(add, sub, mult, div, root, etc)
Ex: 𝑓(𝑥) =
​
3
2
𝑥 +1
TRIGONOMETRIC FUNCTIONS
●​ Convention to use radians unless otherwise stated
1.3 NEW FUNCTIONS FROM OLD FUNCTIONS
TRANSFORMATION
​
Suppose c>0 to obtain the graph of
​
y=f(x)+c, shift y=f(x) c units upward
​
y=f(x)-c, shifts y=f(x) c units downward
COMBINATIONS OF FUNCTIONS
For f(x) and g(x)
-​ f(x)+g(x)=(f+g)(x)
-​ f(x)-g(x)=(f-g)(x)
-​ f(x)*g(x)=fg(x)
-​ f(x) / g(x)=(f/g)(x)
ξ included in
1.5 Cancelation Equations for Sine:
Π
Π
sin(Sin^-1X)=X for − 2 ≤ 𝑥 ≤ 2
sin^-1(sin x)=x for for -1<x<1
Simplify cos(tan-1x) let y tan-1x, then tan y=x
Lets look at sec y
sec2y=tan2y+1
2.1 TANGENT AND VELOCITY PROBLEMS
A Tamhent line touches a curve at a point