Stage 1 Functions and Relations

Stage 1 Functions and Relations

1. Forms to represent a relation

Le tus see the following affirmation: “All number has its doublé”, we could represent the given affirmation as a correspondence among numbers, as it is shown.[pic 1][pic 2]

The relationship described could be represented as ordered pairs:  {(1,2),(2,4),(3,6),(4,8)}

Where the first number of each pair is where the correspondence begins and the second number the correspondence finishes.

The same example can be expressed by table of values, the elements of the first set is called x and second is called y.

The relationship is showing, follows certain “rule”, which each element is corresponding its double, which is expressed as follow:     r: N→ N  … x → 2x

1.1.2 Graphs

The graph of an equation can be drawn when finding enough points to obtain a pattern. Once the graph is drawn, you can decide if the relation is a function or not.

Definition: The domain of a function or relation is the set of all values of the independent variable.

Definition: The range of a function is the set of values of a dependent variable corresponding to all values of an independent variable in the domain.

Definition: Relation is any set of ordered pairs or any correspondence between sets.

The closed dot or black circle (●) indicates ≤,≥, it means, the point belongs to the graph. The open dot (○) indicates <,>, it means, the point doesn’t belong to the graph.

1. Functions in the real world

In situations of the real world, there are almost always, two variable amounts that are related in such a way that the value of one variable depends on the value of the other. For example:

1.- The distance travelled depends on the time elapsed (and also how fast).

Definition: An asymptote is a line that the graph of a function approaches zero.

1. Graph of functions and relations. Criteria of vertical line

You have learned to sketch the equation of a function and observe that all have a common range, for each x-value, you obtain only y-value.

You will know whether are functions, if you draw an imaginary vertical line crosses the figure, if the vertical line crosses the graph, this means for each x-value has at most y-value.

This is called “criteria of vertical line”[pic 3][pic 4][pic 5]

                                 Function

1.2.1 Linear function

Definition: A linear function is a function which general equation is y = mx + b, where m and b are constants and m ≠ 0.

The equation y= mx + b is known as general equation, but if we give concrete values to m and b as y= 3x + 5, then the equation is called particular equation.

If a particular equation m=0, y=7, then y will be equal to a polynomial of zero grade and the equation will be called constant function and not linear function.

1.2.2 Properties of the graph of a linear function

The slope m of a linear function is the ratio  where run is the horizontal distance between two points of the graph and rise is the vertical distance between them.[pic 6]

Formula of slope: If (x1, y1) and (x2, y2) are two points of the graph of a linear function, then:

m= [pic 7]

Definition: Intercepts of a graph, y-intercept of a function is the value of y, when x = 0. The x-intercept of a function is the value of x, when y =0.

Slope-intercept form: If y= mx + b, where m is the slope of the line and b the coordinate of the intersection with the y-axis (the y-intercept or origin coordinate).

Horizontal and vertical lines: *If y= constant, a line with zero slope is a horizontal.

*If x = constant, a line with undefined slope is vertical.

1.2.3 Forms of the linear function or equation of a line.

Forms of the general equation of a linear function:

Y = mx + b        Intercept- slope form: m is the slope and b is the y-intercept.

y-y1 =m(x-xx)   Point-slope form: m is the slope and y (x1,y1) is a point of the line.

Ax + By = C       Standard form: A, B, and C are real numbers.

 +  = 1          Symmetrical form: a is the point where the line crosses the x-axis and b is y-intercept[pic 8][pic 9]

1.2.6 Linear equations and inequalities

Given the mathematical expressions a and b, it happens, they are equal ( a = b ) or are different (a≠b). If they are different, one of the numbers is greater or less than the other one. This relation of “greater than” or “less than” has the symbols >,<, respectively.

Definition: It is called inequality to any expression that refers to the relation between two numbers and therefore has the symbol “>” or “<”. It is called inequation to the inequality which has an unknown quantity.

As symbols we represent these situations as the following:

a is less than b: a <

b is less than a: b < a

a and b are equal: a = b

[pic 10]

• Equation that produce a straight line.        
• Intercept-slope form: y = mx +b
• Point-slope form: y –y1 = m(x – x1)
• Standard form: Ax + By = C
• Intercept or symmetrical form: [pic 11]

1.2..7 Linear inequations and inequalities in a variable

An inequality like x + 4 < 2x is an inequality or lineal inequation because the exponent (implicit) of the variable x is 1.

The set of all real numbers that are solutions of an inequation is called solution set.

To procedures to solve linear inequations in a variable, we use properties of inequalities:

1.- Transitive property:   If a < b and b < c then a < c.

...