Function and Dependency Definition or Description with Formulas
Functions, Function Notation, Dependencies, Formulas
Saying how to obtain a number or quantity from others defines a function, a dependency of one number or variable on others.
What does the equal sign = mean? It means has the same value.
Formulas provide shorthand way to say how one variable y depends on another variable x. Two examples follow.
- Equation y = 3x+1 says how a number y can be computed from x.
So y = f(x) where f(x) = 3x+1. The expression f(x) is read aloud as f at x. - Equation y = x3 says y = h(x) if h(x) = x3 The expression h(x) is read aloud as f at x.
Note: If a function (dependency) is given by a rule or formula q(x) = 4x + 10 then the substitution x = 5 on both sides gives
q(5) = 4(5) + 10 or q(5) = 30.
The expression q(5) is read aloud as q at 5. .Likewise, q at 3 is given by
q(3) = 4(3) + 10 = 12+ 10 = 22
and q at -6 is given by
q(-6) = 4(-6) + 10 = -24+ 10 = -14.
Examples of dependencies and function notation for them follow. A function can be given by an simple or complicated expression.
The letters f, g, h, q used to name or identify functions may be recycled. that is, in different examples or situations, letters may and will represent different computation rules (formulas, dependencies, functions). The old of denoting numbers and quantities, variable or not, will continue.
Here the dependency or function f for obtaining the value of y (the so-called dependent variable) from specified values of others (the so called independent variables) may be described in several ways. Those ways may include formulas, graphs, sets of ordered pairs, tables, arrow diagrams and/or words. This page explores the formula viewpoint.
Think of a function to be a computation or assignment rule that describes how one number depends on others.
Function Notation in Pure and Applied Mathematics
Saying how to compute or obtain a number or quantity from others defines (gives) a dependency, a function.
If the value of a number or quantity y depends on and is unquely determined by the values of numbers a, b, c and so on mathematicians will write y = f(a,b,c, ...) to indicate this dependence where the three dots (ellipsis) stand for the other numbers on which y depends. Another letter except y, a, b and c may be used in place of f in the foregoing function notation y = f(a,b,c, ...) for this dependence.
Note: Physicists and engineers may write y = y(a,b,c, ...) to indicate the same dependence without introducing a new symbol f to denote or name the function or dependence. Mathematicians, pure ones, at least will consider the double occurrence of y in the equation or formula y = y(a,b,c, ...) an over use of the symbol y. Since these are notes for a mathematics course, we will follow the pure mathematics convention in all or most cases.
Note: Mathematics texts and dictionaries may repeat the nonsense that a letter in mathematics denotes a variable, and vice-versa, that a variable in mathematics is given by a letter. But in the formula p = 2pr for the perimeter of a circle, the the letter p stands for a constant and not a variable, while the letter r denotes the radius of a circle. That radius is constant for each circle whose radius is not changing, albeit the radius r may vary between circles. In many formulas for perimeters and areas of co more precise, we may denote number and quantities by single or compound symbols. The numbers and quantities in question may be constant or not, variable or not, in one way or another. There is no guarantee that letter or expression in mathematics denotes a number that varies.
In the pure mathematics usage, the symbol y has the single role of denoting a variable instead of two roles: denoting both a variable and a function - as allowed in the impure usage. In mathematics courses, you should or may have to follow the pure mathematics usage.
For Later Study - Skip on First Reading: The set of points (x,y) for which y = 3x+1 = f(x) is infinite. So all table of values will be incomplete. This set is called the graph of f. With some redundancy,
graph(f) = {(x,y) in R2 | x and y are real numbers with y = 3x + 1}
The graph of the function f(x) = 3x+1 is a straight line.
In the set-based codification of mathematics, we identify a function with its graph. Do not worry about that now.
More Examples From Algebra
Similar to those give above.
- In algebra, we may see formulas y = 2+ 4x to say how y
depends on x and to show how to compute y from the value of x.
So x = 5 gives y = 2 + 4(5) = 2 + 20 = 22. Now we may
rewrite the foregoing if we introduce the function and function
notation f(x) = 2+4x When we give a value for x, the
foregoing formula gives a value for f(x). For instance,
when x = 2, f(x) = 2+ 4x = 2 + 4(5) = 2 + 20 =
22. as before. We are only introducing extra notation.
- In algebra, we may also see formulas y = 3+ 3x + 4x2 to say how y depends on x and to show how to compute y from the value of x. So x = 2 gives y = 3 + 3(2) + 4(2)2 = 3 + 6 +16 = 25. Now we may rewrite the foregoing if we introduce the function and function notation g(x) = 3+ 3x + 4x2 When we give a value for x, the foregoing formula gives a value for g(x). For instance, when x = 2, g(x) = 3+ 3x + 4x2 = 3 + 3(2) + 4(2)2 = 3 + 6 +16 = 25 as before. Here again we are just introducing extra notation, function notation, to describe the dependency of y on x and to give it name g.
Geometric Examples of Functions
Dependent and Independent Variables
Area of Rectangles
The area A of a rectangle is given by its width W times its length L. So we write
This formula shows how to compute area A from the two dimensions L and W of the rectangle. So the value of A depends on or can be computed from the values of L and W. To indicate this dependence of A on the values of L and W we may write
A = L ·W
A = A(L,W) - physicist notation.
or
A = F(L, W) - mathematician notation.
where the value of F(L, W) is given by the product L ·W. That is,
F(L, W) = L ·W
We say that A depends on the values of L and W or that A is function F of L and W.
The values of A, L and W vary or change as we consider different rectangles. So the letters A, L and W are placeholders for three variables, namely the area, the length and the width of a rectangle. We will call the letters A, L and W above variables as they are placeholders for numbers or quantities that vary or may vary between rectangles or examples of rectangles.
We may read F(L,W) aloud as F at the ordered pair or point (L,W). But reading aloud is now get awkward. Where it does, reading in silence or a glance may be preferred. See what is practical - what is not too awkward to read aloud.
Numerical Example A: When W = 4 and L =10, we have
A = F(L, W) = L ·W = 4 ·10 = 40
The replacement of (L,W) by (4, 10) respectively in the formula for F, that is, in
F(L, W) = L ·W
gives
F(4, 10) = 4 ·10 = 40.
Alternatively we may write
F(L, W)| (L,W) = (4,10) = L ·W |(L,W) = (4,10) = 4 ·10 = 40.
- Here we borrow a trick from college mathematics (calculus) we read
F(L, W)| (L,W) = (4,10)
as the expression F(L,W) evaluated at the point where (L,W) = (4, 10).
- We likewise read L ·W| (L,W) = (4,10) as the expression L ·W evaluated at the point where (L,W) = (4, 10).
Numerical Example B: When W = 5 and L = 7, we have
A = F(L, W) = L ·W = 5 ·7 = 35
The replacement of (L,W) by (5, 7), a substitution, in the formula for F, that is, in
F(L, W) = L ·W
gives
35F(5, 7) = 5 ·7 =
So F at the point (5,7) has the value 35 or more briefly, F at (5,7) equals 35. Reading aloud or showing how is not always un-awkward.
Remarks:
- Writing A = F(L, W) says the variable A, its value, depends on those of the two variables L and W; but a second formula
- F(L, W) = L ·W is needed to describes or specifies how.
Instead of writing two lines (1) and (2) separately, we may indicate both in a single line
A = F(L, W) = L ·W
where three parallel lines = indicate equals by definition.
What is a definition: A definition says what something is. A definition is an answer to the question what is.
Summary: the Area of a rectangle A = F(L,W) where L and W denote the dimensions of the rectangle, say L for length and W for Width, and F(L,W) = L*W. Here (L,W) = (4,5) and F(L,W) = L*W gives F(4,5) = 4*5 = 20. That is evaluation of A = F(L,W) when (L,W) = (4,5) gives A = 4*5 = 20. Here the area A of a rectangle is determined by the values of the ordered pair (L,W). All we have done here is introduce function notation and the function concept into the formula A = L*W. The area A is a function of two variables L and W or the point (L,W) in the L-W plane or first quadrant. (blah, blah, blah, ...) Here A = F(L,W) is a real valued function of two real variables or quantities (L,W).
Perimeter of Rectangles
The perimeter P of the rectangle
depends on the values of W and L as well. So following the physicists, we may write P = G(W,L) to indicate this dependence - that P is also a function of the two numbers W and L. Here we say function to indicate the existence of that dependence. The dependence or function G is given by the formula
G(W,L) = 2W + 2L
and by the formula
G(W,L) = 2(W + L)
Describing a single dependence or function in two different ways is acceptable when and only when the different ways all give the same results.
Area of Triangles
In words, the area A of a triangle is given by one half the length B of a base of the triangle multiplied by the height H of the triangle.
We may write still more briefly that the area of a triangle is given by
| A = | 1
2 |
[B ·H] |
To indicate this dependence, we write
A = T(B,H)
where the dependence or function named T is given by
| T(B,H) = | 1
2 |
[B ·H] |
Areas, Perimeters and Diameters of Circles
Let r be our shorthand for the radius of a circle.
Then the area A, perimeter p and diameter d of the circle depend on the radius r. Those dependencies are given or described by
A = f(r) = p r2
p = g(r) = 2p r
and
d = h(r) = 2r
where
f(r) = p r2
g(r) = 2p r
and
h(r) = 2r
The dependencies describes three functions f, g and h - three dependencies.
Summary for Circle Example: the area of a circle is given by A = pr2 or A = f(r) where f(r) = pr2 while the perimeter p = 2pr = g(r) where g(r) = is another function. Here we have two further real-valued functions f(r) and g(r) of a single real variable r. We denote different functions by different symbols f and g. In a classroom when two students share the same name, confusion may results. So different names are best. So when two different functions or dependencies appear in a situation or classroom, use different names or letters.
Function Notation for Transformations
translations, rotations, reflections and dilitations.
Some function notation or formulas to describe four geometric transformations translations, rotations, reflections and dilatations follow. Here one point (X, Y) depends on the value of other point (x,y).
The word transform literally means to change form or at least position. In your earlier studies, you have most used or applied translation, rotation and reflections to change the position of points and triangles. Formulas to describe these transformations or movements follow in function form. Note the use of different letters to denote or name the functions (dependencies).
Translation Example: (X,Y) = f(x,y) where f(x,y) = (x+3, y+4). More generally, secondary II students in Quebec may see prematurely that a translation is given by a formula, dependence or function of the form
(X,Y) = T(a,b)(x,y) = (x+a, y+b) = a function of x, y, a and b.
where a and b are parameters - numbers to be specified or given.
When we give a and b values, like a = 3 and b = 4, the right hand side becomes
f(x,y) = (x+3, y+4) = T(a,b)(x,y) = a
point-valued function of point (x,y). So a function of two variables x and
y comes specifying the value of two variables a and b in a function of four
variables x, y, a and b.
Rotation Thru 90 degrees Example:
(X,Y) = R(x,y) where R(x,y) = (y,-x)
Here the origin of the coordinate system is located at the center and fixed point of the rotation. (In general, if you have a rotation about a center, we can make that center the origin of a coordinate system.)
Reflection Across x-axis Example:
(X,Y) = g(x,y) where g(x,y) = (x, -y)
reverses the sign of the second coordinate, the ordinate.Any horizontal line can serve as the x-axes of a coordinate.
Reflection Across y-axis Example:
(X,Y) = h(x,y) where h(x,y) = (-x, y)
reverses the sign of the first coordinate, the abscissa.Any vertical line can serve as the x-axes of a coordinate.
Reflection Across line y = x Example:
(X,Y) = p(x,y) where p(x,y) = (y, x)
interchanges the first and second coordinates.
You need to investigate or understand why the latter gives a reflection now or later. We may explain why later. Given any line with a 45 degree slope, we can place the origin of many coordinate systems on it.
General Dilatation Examples
Quebec courses in secondary II, III and IV describe these transformations geometrically without emphasizing the formula
(X,Y) = (ax,ay) = Ga(x,y) = a function of x, y and a.
When the value of the variable or parameter a is given, we get a function of two variables (x,y). For example a = 3 gives
(X,Y) = (3x,3y) = G3(x,y) = D(x,y)
For example, D(2,-4) = (3x,3y)|(x,y) = (2, -4) = (3*2,3*(-4)) = (6,-12)
Here we again borrow a trick from college mathematics (calculus) and read
(3x,3y)|(x,y) = (2, -4)
as (3x,3y) evaluated at the point where (x,y) = (2, -4).
The G3(x,y) = (3x,3y) gives G3(x,y) = (0,0). So the origin is a fixed point of the dilatation G3(x,y) = (3x,3y). That being said, each choice of origin and the coordinate axes through gives a coordinate, and the choice can anywhere in the plane. In general, we will assume, if we are given the fixed point of a dilatation, we can put the origin of a coordinate system at the point and then represent the dilatation by a function of the form Ga(x,y) = (ax,ay). How to replace this assumption by a deductive chain of reason is a question for later.
Exercise: Explain how the multiplication of a complex number by a positive real a and the rotation of the result by an angle t can be viewed as multiplication by a complex number whose polar coordinates (r, q) = (a, t)