I. Reminders and complements on numerical functions
1. concept of function
We define a function on a set by associating to each real x of D
a single real denoted by f(x).
- x is the variable;
- f(x) is the image of x by the function f .
- If y = f(x), then x is the antecedent of y by f.
- D is the set of definition of the function, i.e. the set of numbers that have an image by f.
Function defined in ℝ by its literal expression :
Function of two variables :
We call x and z the base and height of a triangle.
Its area is given by the formula: .
Function taking its values in ℕ :
Each real number greater than 1 is assigned the number of divisors of its integer part.
2. Graphical representation of functions
The representative curve of a function f (or graphic representation) is the set of points in the plane with coordinates ( x ; y ) such that :
- the abscissa x is a value of the definition set D;
- the y-intercept is the image of x by f. Therefore y = f(x).
In other words if and only if and y = f(x).
f is the function defined on by .
Draw its representative curve after having completed a table of values.
II. Graphical resolution of equations :
1. Equation of the type f(x) = k (with k a real number)
The solutions of the equation f(x) = k are the abscissae of the points of intersection of the curve and the horizontal line of equation y=k .
On this representative curve of the function f, the equation f(x) = k has for unique solution the number a.
Solving the equation f(x) = g(x), where f and g are two numerical functions, is the same as finding the coordinates of the intersection points of these two curves.
III-Notion of variations on an interval
Consider a function f represented on the interval [-4 ;6] .
A function is said to be increasing over an interval when the images of any two numbers in that interval are always in the same order as the starting numbers. Graphically, the curve “rises”.
A function is said to be decreasing over an interval when the images of any two numbers in that interval are always in the opposite order to the starting numbers. Graphically, the curve “goes down”.
Table of variations :
This is a table where we summarize the variations of the function :
An extremum is any maximum or minimum of the function on its set of definition.
Draw a curve that can represent the function defined on [-2 ; 4] such that :
- f reaches its maximum on [-2 ; 4] in 4;
- The point (-2; 0) is a point on the curve of f;
- An antecedent of 4 is 3 by f;
- The image of 1 is -0.5 by f.
- The minimum of f on [-2 ; 4] is -1, reached in 0;
- f (4) = 5;
- f is decreasing on [-2 ; 0] and increasing on [3,5 ; 4];
- 3 has two antecedents by f which are x = 2 and x = 3.5.
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