Functions and variations : 11th grade math worksheets in PDF.

11th grade math worksheets

Math worksheets in 11th grade on numerical functions and the direction of variation of a function. Calculate the image or antecedent and create the table of variation.

Exercise 1 – Sense of variation of a compound function

Give a decomposition of the function defined by $f(x)\,=\,(x-3)^2\,+2$ that allows to deduce its direction of variation on the interval $I\,=%5D\,-\,\infty\,;\,3%5D$ .

Exercise 2 – Sense of variation

Consider the function defined by $f(x)=x(1-x)\,sur\,\mathbb{R}.$

1. Prove that $f(x)\leq\,\,\frac{1}{4}\,,\gamma\,x\in\mathbb{R}.$

2. Deduce that the function has a maximum in $x=\frac{1}{2}.$

3. Show that $f(x)=\frac{1}{4}-(x-\frac{1}{2})^2$ .

4. Deduce that is increasing on the interval $%5D-\infty;\frac{1}{2}%5B$ and decreasing on $%5D\frac{1}{2};+\infty%5B$ .

Exercise 3 – Comparing two functions

The purpose of this exercise is to compare the two functions f and g defined on $\mathbb{R}$ by :

$f(x)=\frac{1}{1+x^4}\,et\,g(x)=\frac{1}{1+x^2}$

1. Calculate .

2. Deduce the interval on which we have $f\geq\,\,g.$

Exercise 4 – Comparison of functions

The purpose of this exercise is to compare the two functions f and g defined by :

$f(x)=\sqrt{1+x}$ and $g(x)=1+\frac{x}{2}$ on the interval [-1;+\infty[” alt=”[-1;+\infty[” align=”absmiddle” />.

1. Show that $f(x)\geq\,,0$ and $g(x)\geq\,,0$ for all x belonging to $%5B-1;+\infty%5B$ .

2. Calculate and .

3. Prove that $,(f(x),,)^2\leq\,,,(g(x),,)^2$ for all x belonging to $%5B-1;+\infty%5B$ .

4. Deduce a comparison of f and g on the interval $%5B-1;+\infty%5B$ .

5. Draw on the same reference frame the graphical representations of f and g on the interval $%5B-1;+\infty%5B$ .

Exercise 5 – Compound function

Consider the function f defined by $f(x)\,=\,x^2\,-\,1$ on $\mathbb{R}$ .

Give an explicit formula for the function when :

1. $g(x)=\sqrt{x-1}\,sur\,%5B1;+\infty%5B$ .

2. $g(x)=1-\frac{1}{x}\,sur\,\mathbb{R}^*$ .

Exercise 6 – Parity

Study the parity of each of the following functions:
$f(x)=x+\frac{1}{x}\,sur\,\mathbb{R}^*\\g(x)=x^2+\frac{1}{x}\,sur\,\mathbb{R}^*\\h(x)=x+\frac{1}{x^2}\,sur\,\mathbb{R}^*\\k(x)=x^2+\frac{1}{x^2}\,sur\,\mathbb{R}^*$

Exercise 7 – Study of a numerical function

Let $\,f$ be the function defined on $\,\mathbb{R}$ by $\,f(x)=x^2+6x+5$

1. Study the variations of $\,f$ on $\,\mathbb{R}$ .

2. Determine the coordinates of the points of intersection between the representative curve of $\,f$ and the line $\,D$ of equation $\,y=\frac{1}{2}x-2$ .

Exercise 8

Study the variations on $\,\mathbb{R}$ of the function f defined by $\,f(x)=3x-4x^3$ .

Exercise 9

Let f be the function defined on $\,\mathbb{R}$ by :

$\,f(x)=\frac{-4x-4}{x^2+2x+5}$ .

1. Study the variations of f on $\,\mathbb{R}$ .

2. Determine the coordinates of the point A, intersection between the curve representing f and the x-axis.

3. Determine an equation of the tangent T to the representative curve of $\,f$ at point A.

Exercise 10

Study the variations on ]-2 ; 1[ of the function $\,f$ defined by :

$\,f(x)=\frac{-5x^2+4x-8}{x^2+x-2}$ .

Exercise 11 – Canonical and factorized form

Determine the canonical and factorized form of :

$f:x\,\mapsto 2x^2-2(\sqrt{3}-\sqrt{5})x-2\sqrt{15}$

Exercise 12 – Study of functions of the second degree

We note and two polynomial functions of the second degree, defined by :

$f:x,\mapsto ,2x^2,+2x-4$ and $g:x,\mapsto ,-(x+3)(x+2)$

We note and their respective graphical representation in an orthogonal frame $(O,\vec{i},\vec{j})$ .

1. Determine the domain of definition of and then that of .

2. Determine the canonical and then factorized form of .

3. Determine the expanded and then canonical form of .

4. Determine the coordinates of the intersection points between and the axes of the frame.

5. Determine the coordinates of the intersection points between and the axes of the frame.

6. Draw up the table of variation of and then that of .

7. Describe then .

8. Determine the coordinates of the intersection points between and .

9. Study the relative position between the two curves and .

Exercise 13 – Study of an inverse function and the hyperbola

f is the function $x\,\mapsto \,\frac{2}{x}$ defined on $\mathbb{R}^*$ .

g is the function $x\,\mapsto \,-x+3$ defined on $\mathbb{R}$ .

In an orthonormal frame of reference , C and D are the curves representing f and g.

1. Draw the curves C and D.

2. Show that the point of abscissa 1 of D belongs to C.

Find the second point of intersection of these curves.

hint: Check that x² – 3x + 2 = (x – 1)(x – 2)

3. Check the coordinates of these intersection points on the graph.

4. Construct the set of points M(x; y) such that x² y² = 4.

5. A rectangle has area 2 m² and perimeter 6m.

Using the previous graph, find its length and width.

Exercise 14 – Study of a function

Consider the function f defined by :

$f(x)=\frac{(1-x^2)^2}{1+x^2}$

1. Determine its definition set.

2. Prove that f is a positive function on $\mathbb{R}$ .

3. Study the parity of the function f.

4. Draw carefully the graphical representation Cf of the function f.

We will limit ourselves to the interval [- 3 ; 3 ].

5. Give, by graphical reading, the value of the maximum of the function f on :

a. the interval [-1;1].

b. the interval [-2;1].

6. Solve the inequation $f(x)\leq\,,1$ .

Exercise 15 – Comparison of functions
The purpose of this exercise is to compare the functions f and g defined by :

$f(x)=\sqrt{1+x}$ and $g(x)=1+\frac{x}{2}$ on the interval $%5B-1;+\infty%5B$ .

1. Show that $f(x)\geq\,,0$ and $g(x)\geq\,,0$ for all $x\in%5B-1;+\infty%5B$ .

2. Calculate and .

3. show that $(f(x))^2,\leq\,,(g(x))^2$ for all $x\in%5B-1;+\infty%5B$ .

4. Deduce a comparison of f and g on the interval $%5B-1;+\infty%5B$ .

5. Plot on the same coordinate system the graphical representations of f and g on the interval $%5B-1;+\infty%5B$ .

Exercise 16 – Parity

Study the parity of the following functions:

$f(x)=x+\frac{1}{x}$ on $\mathbb{R}^*$

$g(x)=x+\frac{1}{x^2}$ on $\mathbb{R}^*$

Exercise 17 – Compound

Consider the function f defined by on $\mathbb{R}$ .

Give an explicit formula for the function when :

$g(x)=\sqrt{1-x}$ on $%5D-\infty;1%5D$ then $g(x)=1-\frac{1}{x}$ on $\mathbb{R}^*$ .

Exercise 18 – Compound of reference functions

Let the function be defined by $f(x)=\sqrt{4x-1}$ on $I=%5B\frac{1}{4};+\infty%5B.$

Considering the function f as the composite of reference functions, specify the direction of variation of f on the interval I.

Exercise 19 – Sense of variation of a compound function

We give and $g(x)=\frac{1}{x}$ .

We define the function defined on $I=%5D-\infty;\frac{1}{3}%5B$ by .

1. Give the expression of .

2. Determine the direction of variation of on I .

Exercise 20 – Definition set of a compound function

Consider the functions f and g defined by :

$f(x)=x^2-1\,et\,g(x)=\frac{x+1}{x}$ .

1. Calculate .

2. What is the definition set of ?

Exercise 21 – Augmented Function

Let be the function defined by $f(x)=\frac{2x^2}{x^2+3}$ .
1. Determine the real numbers a and b such that $f(x)=a+\frac{b}{x^2+3}$ .

2. Show that f is increased by 2.

Exercise 22 – Canonical form

Consider the function defined on $\mathbb{R}$ by :

1. Determine the canonical form of .

2. Describe the curve of .

Exercise 23 – Draw the curve of the sum of two functions

u and v are shown below.

Draw on this graph the representative curve of the function u + v.

Exercise 24 – Use of a variation table

Here is the table of variations of a function defined on $\mathbb{R}$ :

We give f( – 2) = – 1 and f(2) = 0.
The following functions are defined:

$h:x\,\mapsto \,f(x)+2;r:x\,\mapsto \,f(x+2);p:x\,\mapsto \,f(2x);g:x\,\mapsto \,2f(x)$

1. Give the values of g (1), h (2), p (1) and r ( – 1).

2. Establish the tables of variations of h, r, p and g.

Exercise 25 – Rational function

Let the function be defined by :

$f(x)=\frac{2x+3}{3x+2}$

1. Study the limits of f. Interpret graphically.

2. Study the variations of f. Give the complete table of variations.

3. Determine the possible intersections of (Cf ) with the x-axis.

Exercise 26 – Commutative composite functions

Let and be the functions defined on $\mathbb{R}$ by :

and

Prove that

Exercise 27 – Comparison of roots

Let be in $\mathbb{R}$ .

1. Develop $(\sqrt{a}+\sqrt{b})^2.$

2. Prove that $\sqrt{a}-\sqrt{b}\geq\,\,\sqrt{a+b}.$

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