whatisfunction.m

%% What is a Function?
% *back to* <https://fanwangecon.github.io *Fan*>*'s* <https://fanwangecon.github.io/Math4Econ/ 
% *Intro Math for Econ*>*,*  <https://fanwangecon.github.io/M4Econ/ *Matlab Examples*>*, 
% or* <https://fanwangecon.github.io/MEconTools/ *MEconTools*> *Repositories*
%% 
% *function/mapping*: a mapping (also called a function) is a rule that assigns 
% to every element x of a set X a single element of a set Y. It is written as: 
% 
% $$f:X\rightarrow Y$$ 
% 
% where the arrow indicates mapping, and the letter $f$ symbolically specifies 
% a rule of mapping. When we write:
% 
% $$y=f(x)$$
% 
% we are mapping from argument $x$ in domain $X$ to value $y$ in co-domain Y.
% 
% _*Definitions:*_
%% 
% * *domain*: big $X$ is the domain of $f$
% * *argument*: little $x$ is an element in big $X$, an argument of the function 
% $f$.
% * *co-domain*: big $Y$ is the co-domain of $f$.
% * *image/value*: when $y =f(x)$, we refer to $y$ as the image or value of 
% $x$ under $f$.
% * *range*: $f(X) = \{ y \in Y : y = f(x) \text{ for some } x \in X  \}$
% * *graph:* "The graph of a function of one variables consists of all points 
% in the Cartesian plane whose coordinates (x,y) satisfy the equation y = f(x)" 
% (SB)
%% 
% In some textbooks, $x$ is called independent or exogenous variables, and $y$ 
% is called dependent or endogenous variables. We will avoid using those words 
% to avoid confusion.
%% 
% _*This is a function*_:

figure();
x = 0:pi/100:2*pi;
y = sin(x);
plot(x,y);
grid on;
%% 
% _*This is NOT a function*_:

figure();
x = 1; y=1; r=1;
th = 0:pi/50:2*pi;
xunit = r * cos(th) + x;
yunit = r * sin(th) + y;
h = plot(xunit, yunit);
grid on;
%% 
% *A Linear Function*
% 
% A linear function, polynomial of degreee 1, has slope $m$ and intercept $b$. 
% Linear functions have a constant slope.

figure();
m = 0.5;
b = 1;
ar_x = linspace(-5, 10, 100);
ar_y = ar_x*m + b;
h = plot(ar_x, ar_y);
% Title
title({['Linear function with slope m=' num2str(m) ' and y-intercept=' num2str(b)]});
% axis lines
xline0 = xline(0);
xline0.HandleVisibility = 'off';
yline0 = yline(0);
yline0.HandleVisibility = 'off';
grid on;
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