1 What is a Function?

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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:

\[f:X\to Y\]

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: \(f(X)=\lbrace y\in Y:y=f(x)\;\textrm{for}\;\textrm{some}\;x\in X\rbrace\)

  • 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;