Download generalities on functions - Lycée Hilaire de Chardonnet

GENERALITIES ON FUNCTIONS
I ) DEFINITIONS OF FUNCTIONS
The set D of all possible input values is called the domain of the function. The set of all values of ƒ(x) as x varies
throughout D is called the range of the function. The range may not include every element in the set Y. The domain and range
of a function can be any sets of objects, but often in calculus they are sets of real numbers. One can use even or odd
functions.
Another way to visualize a function is its graph. If ƒ is a function with domain D,
its graph consists of the points in the Cartesian plane whose coordinates are the inputoutput pairs for ƒ. We can find relative maximums or relative minimums. If the
graph of a function climbs or rises as you move from left to right, we say that the function
is increasing. If the graph descends or falls as you move from left to right, the function
is decreasing.
Sometimes a function is described by using different formulas on different parts
of its domain : this may be a piecewise-defined function. One example is the
absolute value function, another is (on the left) the floor function. Another example : if
x < 0 then f ( x) = − x , if 0 ≤ x ≤ 1 then f ( x) = x ² , otherwise f ( x) = 1 .
II) SOME PARTICULAR FUNCTIONS
Excepted the linear functions, there are a number of important types of functions frequently encountered in calculus.
y = x n ), where n is an integer (different cases, if n is positive or not).
n
2
1
Now introduce the polynomial function: f ( x ) = an x + ... + a2 x + a1 x + a0 , where n is a nonnegative integer
and the numbers are real constants (called the coefficients of the polynomial). . If the leading coefficient an ≠ 0 and n > 0 ,
These graphs are from power functions (
then n is called the degree of the polynomial. Linear functions (excepted 0) are polynomials of degree 1. Polynomials of
degree 2, usually written as are called quadratic functions. Likewise, cubic functions are polynomials of degree 3. A
rational function is a quotient or ratio of two polynomials.
More generally, an algebraic function is a function constructed from polynomials using algebraic operations
(addition, subtraction, multiplication, division, and taking roots). Rational functions are special cases of algebraic functions.
Trigonometric functions are constructed from sine and cosine functions. There are also functions of the form
f ( x) = a x called exponential functions, and their inverse f ( x) = log a ( x) called logarithmic functions (in Tle Class)
III) OPERATIONS
To shift the graph of a function straight up, add a positive constant to the right-hand side of the formula
y = f ( x)
To scale the graph of a function y = f ( x) is to stretch or compress it, vertically or horizontally. This is
accomplished by multiplying the function ƒ, or the independent variable x, by an appropriate constant.
Shifting and scaling are operations on functions, like addition and multiplication.
If ƒ and g are functions, we can work on the composite function (“ƒ composed with g”) is defined by
( fog )( x) = f ( g ( x)) . The domain of fog consists of the numbers x in the domain of g for which g(x) lies in the domain
of ƒ. For instance, if
f ( x) = x ² + 1 and g ( x) = x , fog ( x) = ( x )² + 1 = x + 1 , but only defined on [ 0; +∞[