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QM1 Mathematics Sowiso theoryChapter 1: functions of one variable, part 1
Introduction to functions
- The notion of function
Function rule of this function is f(x) = x^2, or y = x^2. The real number y that the
function f assigns to a number x is denoted by f(x) and is called the value of (the
function) f at x or the function value at x. This value is also called the image of x
under f.
The set of all real numbers that can be entered in the function is called the domain of
the function. The domain of the function g with function rule g(x) = 1/x cannot contain
the number 0, since it is not possible to divide by 0. We then say that g is not defined
at 0.
We call x (the argument) the independent variable an y the dependent variable.
Other variables can occur in the function rule. For instance: f(x) = a X x^2 + t. Here, a
and t play the role of constants; these variables are called parameters.
- Arithmetic operations for functions
Let f and g be real valued functions with the same domain and let a be a real number.
We define:
The sum function f + g as the function with rule (f + g)(x) = f(x) + g(x).
The scaled function a X f as the function with rule (a X f)(x) = a X f(x).
The difference function f – g as the function f + (-1) X g.
The product function f X g as the function with rule (f X g)(x) = f(x) X g(x).
The quotient function f/g as the function with rule f/g(x) = f(x)/g(x).
Because all five are determined by arithmetic operations on values of the corresponding
function, these operations are called arithmetic.
- The range of function
The set of all values f(x) for x in the domain of f is called the range of f.
If p is a real number in the domain of f with f(p) = 0, then p is called a zero of f.
The range of a function depends on the domain of the function: the larger we choose the
domain, the larger in general the range will be.
- Functions and graphs
We often work with x,y-plane, whose elements are the pairs [x,y] of real numbers x and
y. In this plane we can visualize functions. A graph is a set of points of the plane.
A function f can be made visible by drawing a graph: each value of x from the domain f
corresponds to a unique value for y, being f(x). The set of point [x,y] thus obtained, is
the graph of the function f.
Lines and linear functions
- Linear equations with a single unknown
A linear equation with unknown x is an equation that has form: a X x + b = 0, in
which a and b are real numbers.
Solving the equation is finding all values of x for which the equation is true. Such a
value is called a solution of the equation. The values of x for which the equation is true,
form the solution of the equation, also called the solution set.
Equations with x us unknown are called equations in x.
,The expression to the left of the equal sign (=) is called the left-hand side of the
equation (for the equation above this is a X x + b), and the expression on the right of it
is the right hand side (for the equation above this is 0).
The expressions a X x and b in the left hand side are called terms. Because b and 0
occur without x, they are called constant terms, or simply constants. The number a is
called coefficient of x.
- The general solution of a linear equation
in general the solutions of the equation a X x + b = 0 with unknown x can be found as
follows:
Case Solution
A≠0 Exactly one: x = -b/a
A = 0 and b ≠ 0 No real solution
A = 0 and b = 0 True for all real values of x
- The equation of a line
The solution to the equation a X x + b X y + c = 0 can be drawn in the plane. They are
the points [x,y] satisfying a X x + b X y + c = 0. If a ≠ 0 or b ≠ 0, then these points
form a straight line, or simply just a line.
If b ≠ 0, then the equation can be written as y = -a/b x – c/d. For, these are the
solutions if we consider x as a parameter and y as unknown. This indicated that
for every value of x there is a point [x,y] with y equal to -a/b x – c/d.
* If a ≠ 0, the line is oblique (by oblique we mean neither horizontal neither
vertical).
* If a ≠ 0, then the value of y is constant, equal to -c/b. In this case the line is
horizontal.
In the exceptional case b = 0 the equation looks like ax + c = 0.
* If a ≠ 0, then the line is vertical.
* If a = 0 (c ≠ 0, then there are no solutions, c = 0, then each pair of values of
[x,y] is a solution).
A straight line can be described in different ways.
1. The solutions [x,y] to an equation a X x + b X y + c = 0 with unknowns x and y.
Here a, b and c are real numbers such that a and b are not both equal to zero.
2. The line through two given points in the plane; if P = [p, q] and Q = [s, t] are
points in the plane, then the line though P and Q has equation a X x + b X y + c
= 0 with a = q – t, b = s – p, and c = t X p – Q X s.
3. The line through a given point, the base point, and a direction, indicated by the
number -a/b, where a and b are as in the equation given above; this number is
called the slope of the line.
4. The line with function representation y = p X x + q if b ≠ 0 and x = r otherwise;
here we have p = -a/b (the slope), q = -c/b (the intercept), which is the value
of y for x = 0 and r = -c/a in terms of the above a, b, and c. This can be seen as
a special case of the previous description, with base point [0, q]. In the case
where b ≠ 0, the variable y is a function of x, in the other case, x is a constant
function of y.
- Equations and lines
Two lines in the plane can be in three different positions relative to each other:
They can cross (or: intersect in one point)
They can be parallel but unequal, or
They can be identical
The equations of these lines with unknowns x and y have, in the three respective cases:
one solution (regular), no solution (contradicting), and the same line as solution
(dependent).
, In all cases, the points which lie on both lines are the solution to the system of two
equations. The first case occurs then and only then if the slopes of these two lines are
different.
Quadratic functions
A quadratic function f is given by f(x) = ax^2 + bx + c with a, b and c being real
numbers and a ≠ 0.
The graph associated with such a function is called a parabola.
When a > 0 then we speak of a parabola opening upwards and when a < 0 of a
parabola opening downwards.
To find the zeros of a quadratic function, we need to find the points of intersection of the
parabola with the x-axis. We can do this in three ways:
Completing the square
Abc-formula
Factorization
- Completing the square
Completing the square is rewriting a quadratic expression in x as an expression in which
x only occurs once, in the base of a second power. To be precise, if a, b and c are real
numbers then:
With this method we cannot only solve quadratic equations, but also determine what the
top of a parabola is:
The quadratic polynomial ax^2 + bx + c in which a ≠ 0, can be written as:
If a > 0, then the extreme is the lowest point of the parabola opening upwards.
If a < 0, then the extreme is the highest point of the parabola opening
downwards.
In other words, the quadratic function ax^2 + bx + c in x has a minimum or maximum
(depending on a > 0 or a < 0) for
- The quadratic formula
By completing the square, we can rewrite the quadratic expression ax^2 + bx + c as
Let a, b, and c be real numbers with a ≠ 0. The equation ax^2 + bx + c = 0 can be
reduced to
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