per 2 Weinstein, Cheyenne

182 days ago by cheyweinstein

Domain: the set of all possible input values (in our case~ x), which allows the function formula to work

Range: the set of all possible output values (in our case~ f(x)), which result from using the function formula

Function: a set of mathematical operations performed on one or more inputs (variables) that results in an output

Let us start with something simple~! like...
f(x)= x^2 
       

We will now continue to plug the integers between -10 and 10 into the function to demonstrate how f(x) and x may relate to eachother for the function above
[(x,f(x)) for x in [-10..10]] 
        
[(-10, 100), (-9, 81), (-8, 64), (-7, 49), (-6, 36), (-5, 25), (-4, 16),
(-3, 9), (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4), (3, 9), (4, 16), (5,
25), (6, 36), (7, 49), (8, 64), (9, 81), (10, 100)]
[(-10, 100), (-9, 81), (-8, 64), (-7, 49), (-6, 36), (-5, 25), (-4, 16), (-3, 9), (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4), (3, 9), (4, 16), (5, 25), (6, 36), (7, 49), (8, 64), (9, 81), (10, 100)]

Since the domain of a function is all the possible input values, and the range is all possible output values, we can now plot the number above into graph form
plot (f,-10,10) 
       
points ([(x, f(x)) for x in [-10..10]]) 
       

Since we have now described a function, we can now move on to function composition :DDDD


Function Composition: the application of one function to the results of another

Let us now have a function with the domain of x and the range of g(x)...
g(x)=(2*x)+ 12 
       

We will now create a composite function by creating((g * f)(x)) or by g(f(x))

g(f(x)) 
        
2*x^2 + 12
2*x^2 + 12

We can also create the composite function f(g(x))

The functions g and f are said to commute with each other if g ∘ f = f ∘ g; however, in general, composition of functions will not be commutative

f(g(x)) 
        
4*(x + 6)^2
4*(x + 6)^2

It is time for us to move onto inverse functions

Inverse Function: function that undoes another function that is invertible -- the inverse function is then uniquely determined by ƒ and is denoted by ƒ−1

Let us start with a new, simple function to demonstrate the idea of an inverse function
f(x)= 3*x - 7 
       

We can then let f(x) become y
y = 3*x - 7 
       

From there, we must switch y for x and x for y within the equation and solve~
x = 3*y - 7 
       
(x/3)+(7/3)= y 
        
Traceback (click to the left of this block for traceback)
...
SyntaxError: can't assign to operator
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_48.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("KHgvMykrKDcvMyk9IHk="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmpt9rafM/___code___.py", line 3
    (x/_sage_const_3 )+(_sage_const_7 /_sage_const_3 )= y
SyntaxError: can't assign to operator
y = (x+7)/3 
       

There is our demonstration of an inverse function and how to solve them~~~~~~~~~~~~~~

Basically, we have now covered the topics of "domain, range, composition, and inverse." Until next time!