Functions
qualitative VS quantitative.
3 ingredients of a function: a set of inputs (domain), a set of outputs (range), and a rule to assign exact one output to every input (the rule).
If 
are two functions, the sum, the difference, the product, the quotient and the composition of 
and 
.
#card=math&code=f_1%28x%29): the value of at 
.
is define by the rule that the value of this function at is equal to %2Bf_2(x)#card=math&code=f_1%28x%29%2Bf_2%28x%29). Analogously for the $-, \times, $ and 
.
We suppose the domains of are 
. Then has domain $A_1 \cap A_2 $. So are the difference, the product and the quotient. Moreover, 
, should be defined at those points inside 
such that is not being zero.
The composition of and : Say $f_1 $ has domain 
and range 
, has domain 
and 
. Suppose in addition that is a subset of . Then we can define the composition 
. It is d1efined as following: 
, then %20%5Cin%20B_1#card=math&code=f_1%28x%29%20%5Cin%20B_1) . And we regard #card=math&code=f_1%28x%29) as a member of . So that we can talk about the value of at the point #card=math&code=f_1%28x%29). This value is the value of at 
.
Five types of functions:
algebraic (polynomials), exponential and their inverse logarithmatic, trigonometrical and their inverse. These are called “basic elementary functions”.
Elementary functions: the 
and 
of any basic elementary functions.
The good thing: elementary functions on their respective domains, are continuous, differentiable.
Sketch the Graph of a function
Think about the graph of 
.
Determine the domain. (also determine the size/scale of your picture.)
Is the function an even/odd function? Is the function a periodic function?
(If “yes”, can copy and hence save work)
Determine the monotonicity of the given function. (using derivative/differentiation)
Determine the (local) minima and maxima. (using derivative/differentiation)
Is there any asymptotic line for the function?
Determine the values at certain special points (for example, 
%2C%20f(1)%2C%20f(2)#card=math&code=f%280%29%2C%20f%281%29%2C%20f%282%29), or when is the function being zero).
Sketch the graph.
More questions:
We recall the graphs of
, the notion of asymptotic lines
, the notion of inflection points. (the point where the function changes its convexity)
, 
#card=math&code=3x%5E2-2x%3Dx%283x-2%29). At 
the derivative being zero, so these two points are mimimum and maximum points.
To know more about a function, quantitative information is required. Such as the monotonicity, minima and maxima.
Example: 
, 
%20%3D%202#card=math&code=%5Clim%7Bx%5Cto%200.5%7D%20%281%2Fx%29%20%3D%202). %3D0#card=math&code=%5Clim_%7Bx%20%5Cto%20%2B%5Cinfty%7D%20%281%2Fx%29%3D0). We regard 
is the “largest” point. So can only approach to on the left.
%20%3D%20-0%3D0#card=math&code=%5Clim_%7Bx%20%5Cto%20-%5Cinfty%7D%20%281%2Fx%29%20%3D%20-0%3D0).
The notion of asymptotic line of a function.
Two cases: vertical or non-vertical.
Vertical asymptotic line: If there is point 
, so that when approahches 
either on the left or right side, the limit of 
#card=math&code=f%28x%29) is 
, then we say 
is a vertical asymptotic line for #card=math&code=f%28x%29).
Non-vertical asymptotic line 
. If the limit 
-(ax%2Bb)%5D%3D0#card=math&code=%5Clim%7Bx%20%5Cto%20%2B%5Cinfty%7D%20%5Bf%28x%29-%28ax%2Bb%29%5D%3D0) or -(ax%2Bb)%5D%3D0#card=math&code=%5Clim_%7Bx%20%5Cto%20-%5Cinfty%7D%20%5Bf%28x%29-%28ax%2Bb%29%5D%3D0), then we say is an asymptotic line for #card=math&code=f%28x%29).
