# Functions exercises | Numerical functions

## Numerical function exercise 1.

I/ g is a numerical function defined on ℝ -{1} by:  g(x)=x²-2x+ln |x-1|

1-Study the change of the function g And calculate g(0) and g(2).

2-Conclude the sign of g(x) According to the value of x.

II/  f is a numerical function defined on ℝ -{1} by

and (c)  Its graphic representation in a homogeneous and orthogonal coordinate system  (o;i,j) .

1-Prove that for every x of  ℝ -{1} Then:

2-
Prove that the curve (c) accepts two asymptotic lines, one of which is oblique (Δ), Then write an equation of (Δ).
3-Study the position of the curve (c) in relation to the straight line (Δ).

4-Prove that the curve (c) accepts two tangents (T) and (T') with the parallel of a straight line (Δ), then write the equations of  (T) and (T').

5- Prove that the point w(-1;1) is the center of symmetry of the curve (c).
6- Prove that the curve (c) accept two inflection point, And select them.
7-Create the two tangents (T) and (T') and the curve (c).
8-  function defined on ℝ* by:

and (Ch) Its graphic representation in the previous coordinate system.
prove that (Ch) is the image of (C) by Translation. And draw it.

## Numerical function exercise 2.

g is a function defined on ]0;+∞[  by: g(x)=x²-2+lnx

### 1- Study the change of the function g on ]0;+∞[ .

2-Prove that the equation accepts a single solution α Where:
1.31   < a <  1.32

3-Conclude the sign of g(x) on  ]0;+∞[ .

II)  We consider the function f defined on ]0;+∞[  by:
f(x)=x²+(2-lnx)²

1-Express the derivative function f in terms of g(x)

2- conclude the change of the function f  on  ]0;+∞[ .

## Numerical function exercise 3.

### f is a Polynomial function where:

1-Calculate f(-1) and then find the real numbers a , b, c  where:

f(x)=(x-1)(ax+bx+c)

2-numerical function defined on by:

and (D) straight line its equation y=2x+2
a) Prove that the curve (Cg) Representative of the function g and the straight line (D) They have a common point A its range is zero.
b) Calculate g'(x) and write the equation of the tangent of (Cg) at the point A.

objectives:

### The objective of the second exercise:Study the change of the function; Define an restriction to solve the function; determination its sign of the function; and conclude the change of the function from another function.The objective of the third exercise:analysis Polynomial  to two factors; and determine the point of intersection of a curved with a straight line.

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