High school math exercises

Exercise example N°1438 :

Consider the expression `E=(8*x+4)^2-(8*x+4)*(7*x-5)`.
1. Expand and reduce E.
2. Factor E.
3. Solve the equation `(9+x)*(4+8*x)=0`.

mathematics exams and competitions expansion of algebraic expressions solving equations and inequations of the first degree factoring equations 9th Grade 10th Grade expand_and_simplify

The purpose of this exercise is to practice factoring, expanding, simplifying algebraic expression and solving equation.

Exercise example N°1439 :

Consider the expression `E=(2*x+10)^2-(2*x+10)*(5*x-4)`.
1. Expand and reduce E.
2. Factor E.
3. Solve the equation `(14-3*x)*(10+2*x)=0`.

mathematics exams and competitions expansion of algebraic expressions solving equations and inequations of the first degree factoring equations 9th Grade 10th Grade expand_and_simplify

The purpose of this exercise is to practice factoring, expanding, simplifying algebraic expression and solving equation.

Exercise example N°1501 :

Solve the following equation: `x/2-3=0`.

solving equations equations 10th Grade equation_solver

This corrected exercise allows to practice solving linear equations with one unknown of the form ax+b=0.

Exercise example N°1502 :

Solve the following equation: `z^2+1-2*z=0`.

solving equations equations 10th Grade equation_solver

The purpose of this exercise on quadratic equations is to practice solving 2nd degree equations and null product equations.

Exercise example N°1503 :

Solve the following equation: `y^2-16=0`.

solving equations equations 10th Grade equation_solver

The purpose of this exercise is to solve a second degree equation by reducing it to solving a first degree equation.

Exercise example N°1504 :

Solve the following equation: `z^2+1-2*z=0`.

solving equations equations 10th Grade equation_solver

The goal of this exercise is to solve a null product equation of the type a*b=0, with a=0 or b=0.

Exercise example N°1505 :

Specify if the function `f:x->7-3*x^2` is even, odd, neither even nor odd.

square and inverse functions functions 10th Grade is_odd_or_even_function

The purpose of this corrected exercise is to determine the parity of a function (specify whether the function is even or odd).

Exercise example N°1506 :

With the help of the graphical representation of the function shown below in an orthogonal reference, indicate if the function is even, odd, neither even nor odd.

square and inverse functions functions 10th Grade is_odd_or_even_function

The purpose of this corrected exercise is to determine graphically the parity of a function (specify whether the function is even or odd).

Exercise example N°1507 :

To which type of curve corresponds the following plot ?

square and inverse functions functions 10th Grade

The aim of this corrected exercise is to recognize from their graphical representations the square and inverse functions.

Exercise example N°1508 :

The representative curve of the function f is given below. Find graphically one or more integer values of x on the interval [-5,5[ which verify the equation f(x)=1. You can use the red cursor to read the coordinates of the points.

solving equations square and inverse functions equations 10th Grade equation_solver

The purpose of this exercise is to solve graphically an equation.

Exercise example N°1509 :

Convert to degrees `pi/3` radians.

sine and cosine functions functions 10th Grade

The goal of this math exercise is to convert angles expressed in degrees into radians.

Exercise example N°1510 :

The angles are expressed in radians. Give the exact value of the following expression `pi/3`

sine and cosine functions functions 10th Grade simplify_trig

The aim of this math exercise is to calculate expressions that contain sines, cosines and remarkable angles.

Exercise example N°1511 :

Calculate the absolute value of `C=8+9`.

order, absolute value, inequations functions Numbers 10th Grade abs

This corrected exercise consists simply in calculating the absolute value of a numerical expression.

Exercise example N°1512 :

Calculate the absolute value of `F=2/3-3/7`.

order, absolute value, inequations functions Numbers 10th Grade abs

This corrected exercise consists simply in calculating the absolute value of an algebraic expression composed of fractions.

Exercise example N°1513 :

Solve the following equation `|x-4|=2`.

order, absolute value, inequations equations functions 10th Grade equation_solver

The purpose of this corrected exercise is to solve an equation with an absolute value (equation of the form |x-a|=b).

Exercise example N°1514 :

Solve the following equation `|x+9/2|=9/4`.

order, absolute value, inequations equations functions 10th Grade equation_solver

The purpose of this corrected exercise is to solve an equation with an absolute value (equation of the form |x-a|=b).

Exercise example N°1515 :

Indicate by which number the "question mark" must be replaced in the prime decomposition of 60 so that the following equality is verified.
60 = 3*5*?*?

order, absolute value, inequations Numbers 10th Grade prime_factorization

The purpose of this corrected exercise is to complete the decomposition of a number into prime numbers.

Exercise example N°1516 :

Give the decomposition of 854 into a product of prime numbers by ordering the factors and using the power operator ^ if necessary.

order, absolute value, inequations Numbers 10th Grade prime_factorization

The purpose of this exercise is to find the ordered decomposition of a number into primes.

Exercise example N°1517 :

51 is an integer, is it prime ?

order, absolute value, inequations Numbers 10th Grade prime_factorization

The purpose of this corrected arithmetic exercise is to determine if a number is a prime number.

Exercise example N°1518 :

Give the product decomposition of the following expression 30*16 by ordering the factors and using the power operator ^ if necessary.

order, absolute value, inequations Numbers 10th Grade prime_factorization

The goal of this exercise is to find the ordered decomposition of a product of numbers into primes.

Exercise example N°1520 :

Write in the form of an irreducible fraction the following fraction `(40*28)/(35*24)` using the decomposition into prime factors.

fractions Numbers integers and rational numbers 10th Grade fraction

The purpose of this exercise is to simplify a fraction using the decomposition of a number into a product of prime factors.

Exercise example N°1524 :

Find the ordinate of the direction vector of the line whose equation is `y=-7/10*x+6` which has abscissa 1

vectors equations equations of lines and linear systems 10th Grade fraction

The objective of this exercise is to determine the ordinate of a direction vector from a line equation.

Exercise example N°1539 :

Put into irreducible fraction form: `((-9)/(20))/((-36)/(-15))`.

fractions Numbers 10th Grade fraction

The purpose of this exercise is to use algebraic computation techniques to determine the irreducible form of a division of fractions.

Exercise example N°1541 :

Put into irreducible fraction form: `((-9)/(20))/((-36)/(-15))`.

fractions Numbers 10th Grade fraction

The purpose of this corrected calculus exercise is to use algebraic calculus techniques to simplify a product of fractions.

Exercise example N°1601 :

Compute the discriminant of the following polynomial: `2*x^2+4*x`.

second degree polynomials solving equations 11th Grade discriminant

The goal of this corrected exercise is to calculate the discriminant of a second degree polynomial from its algebraic form.

Exercise example N°1602 :

How many solutions does the following equation have: `2*x^2-x` ?

equations second degree polynomials solving equations 11th Grade discriminant

The purpose of this corrected exercise is to find the number of solution of a second degree equation as a function of the discriminant.

Exercise example N°1603 :

Give the roots of the following equation `4*x^2+x-2`

equations second degree polynomials solving equations 11th Grade equation_solver

The purpose of this corrected exercise is to use the discriminant of a second degree equation to find its roots.

Exercise example N°1604 :

Calculate the derivative number of the function f(x) = `2+2*x^2` at point a = -2

derivatives of functions functions 11th Grade derivative

The aim of this corrected maths exercise is to calculate the derivative number of a function.

Exercise example N°1605 :

Let f be the function defined by f(x)= `-x-2*x^2+x^3` , calculate the derivative of f, `f'(x)`.

derivatives of functions functions 11th Grade derivative

The purpose of this exercise is to determine through the methods of algebraic calculations the derivative of a polynomial function.

Exercise example N°1606 :

Let f be the function defined by f(x)= `2*sqrt(x)` , calculate the derivative of f, `f'(x)`.

derivatives of functions functions 11th Grade derivative

The purpose of this corrected math exercise is to calculate the derivative of a square root.

Exercise example N°1607 :

Let f be the function defined by f(x)= `1/(3*x^2)` , calculate the derivative of f, `f'(x)`.

derivatives of functions functions 11th Grade derivative

The purpose of this corrected math exercise is to calculate the derivative of a quotient.

Exercise example N°1608 :

Let f be the function defined by f(x)= `1/(4-2*x+x^2)` , calculate the derivative of f, `f'(x)`.

derivatives of functions functions 11th Grade derivative

The purpose of this corrected math exercise is to calculate the derivative of a quotient and a polynomial.

Exercise example N°1609 :

Let f be the function defined by f(x)= `-3-3*x+2*x^2+x^3-5*sqrt(x)` , calculate the derivative of f, `f'(x)`.

derivatives of functions functions 11th Grade derivative

The goal of this corrected math exercise is to calculate the derivative of a polynomial and a square root.

Exercise example N°1610 :

Let f be the function defined by f(x)= `sqrt(3*x)` , calculate the derivative of f, `f'(x)`.

derivatives of functions functions 11th Grade derivative

The purpose of this corrected math exercise is to calculate the derivative of a function composed of a square root and a polynomial.

Exercise example N°1611 :

Let f be the function defined by f(x)= `(4+2*x)/(1-4*x)` , calculate the derivative of f, `f'(x)`.

derivatives of functions functions 11th Grade derivative

The goal of this exercise on functions is to calculate the derivative of a quotient of polynomials.

Exercise example N°1612 :

Let f be the function defined by f(x)= `4*sqrt(x)*(1+2*x)` , calculate the derivative of f, `f'(x)`.

derivatives of functions functions 11th Grade derivative

The purpose of this exercise on functions is to calculate the derivative of the product of a square root and a polynomial.

Exercise example N°1613 :

Let f be the function defined by f(x) = `5*x^2-2*x-4`.
1. Calculate the derivative of the function f at the point of abscissa -2.
2. Deduce an equation of the tangent to the curve representing the function f at the point of abscissa -2.

derivatives of functions functions 11th Grade equation_tangent_line

The purpose of this corrected math exercise is to calculate the derivative number of a function and derive the equation of a tangent to a curve.

Exercise example N°1614 :

Let the sequence (`u_(n)`) be defined for any natural number n by `u_(n)=(-5-4*n)/(4+3*n)`.
1. Compute `u_(0)`
2. Compute `u_(1)`

numerical sequences 11th Grade sequence

The purpose of this exercise on numerical sequences is to calculate terms of a sequence defined from a rational fraction function.

Exercise example N°1615 :

Let the sequence (`u_(n)`) be defined for any natural number n by `u_(n)=-4-4*n`.
1. Compute `u_(3)`
2. Compute `u_(7)`

numerical sequences 11th Grade sequence

The purpose of this exercise on numerical sequences is to calculate terms of a sequence defined by a linear function.

Exercise example N°1616 :

Let the sequence (`u_(n)`) be defined for any natural number n by `u_(n)=(-1)^n*4^(n+1)`.
1. Compute `u_(1)`
2. Compute `u_(2)`

numerical sequences 11th Grade sequence

The purpose of this exercise on numerical sequences is to calculate terms of a sequence defined by a power function.

Exercise example N°1617 :

Let the sequence (`u_(n)`) be defined for any natural number n by `u_(n)=sqrt(3+3*n)/(5+3*n)`.
1. Compute `u_(4)`
2. Compute `u_(6)`

numerical sequences 11th Grade sequence

The purpose of this exercise on numerical sequences is to calculate terms of a sequence defined from a fraction and a square root.

Exercise example N°1618 :

Let the sequence (`u_(n)`) be defined for any natural number n by `u_(0)= 2 ` and `u_(n+1)` = `1+u_(n)`.
1. Compute `u_(3)`
2. Compute `u_(5)`

numerical sequences 11th Grade recursive_sequence

The purpose of this exercise on numerical sequences is to calculate terms of a sequence defined by recurrence with a linear function.

Exercise example N°1619 :

Let the sequence (`u_(n)`) be defined for any natural number n by `u_(0)= 2 ` and `u_(n+1)` = `-2+2*u_(n)^2`.
1. Compute `u_(2)`
2. Compute `u_(4)`

numerical sequences 11th Grade recursive_sequence

The purpose of this exercise on numerical sequences is to calculate terms of a sequence defined by recurrence with a quadratic function.

Exercise example N°1620 :

Let the sequence (`u_(n)`) defined by `u_(n)` = `(2+n)/(2+5*n)`.

Express as a function of n the terms of `u_(n+3)`.

numerical sequences 11th Grade 12th Grade

The purpose of this exercise on numerical sequences is to write in algebraic form one of the terms of the sequence.

Exercise example N°1621 :

Let the sequence (`u_(n)`) defined by `u_(n)` = `-3-3*n`.

Express as a function of n the terms of `u_(n+1)`.

numerical sequences 11th Grade 12th Grade

The purpose of this exercise on numerical sequences is to write in algebraic form one of the terms of the sequence.

Exercise example N°1622 :

Let the sequence (`u_(n)`) be defined for any natural number n by `u_(0)= 3 ` and `u_(n+1)` = `-3+u_(n)`.
Is this sequence increasing or decreasing?

numerical sequences 11th Grade 12th Grade

Exercise on the direction of variation of a simple numerical sequence: constant sequences, increasing sequences and decreasing sequences.

Exercise example N°1623 :

Let the sequence (`u_(n)`) be defined for any natural number n by `u_(0)= 4 ` and `u_(n+1)` = `u_(n)/5`.
Is this sequence increasing or decreasing?

numerical sequences 11th Grade 12th Grade

Exercise on the direction of variation of a numerical sequence with a fraction: constant, increasing and decreasing sequences.

Exercise example N°1624 :

Let the sequence (`u_(n)`) defined for any natural number n by `u_(0)= -3 ` and `u_(n+1)` = `-7+u_(n)`.

1. Is (`u_(n)`) an arithmetic or a geometric sequence ?
2. What is the reason of (`u_(n)`)
3. Give the expression of `u_(n)` as a function of n.

numerical sequences 11th Grade 12th Grade

Exercise on arithmetic sequences, on geometric sequences and on common difference and on common ratio.

Exercise example N°1625 :

Let the sequence (`u_(n)`) defined for any natural number n by `u_(0)= -1 ` and `u_(n+1)` = `-9*u_(n)`.

1. Is (`u_(n)`) an arithmetic or a geometric sequence?
2. What is the reason of (`u_(n)`).
3. Give the expression of `u_(n)` as a function of n.

numerical sequences 11th Grade 12th Grade

Exercise on geometric sequences, on arithmetic sequences and their reason.

Exercise example N°1626 :

Let (`u_(n)`) be an arithmetic sequence of common difference -6, and of first term `u_(0)= 1 `.

1. Give the expression of `u_(n)` as a function of n.
2. Compute `u_(3)`

numerical sequences 11th Grade 12th Grade

This exercise allows you to practice the calculation of the terms of an arithmetic sequence from its common difference and its first term.

Exercise example N°1627 :

"Let (`u_(n)`) be a geometric sequence of reason 8, and of first term `u_(0)= 2 `.
1. Give the expression of `u_(n)` as a function of n
.
2. Compute `u_(5)`.

numerical sequences 11th Grade 12th Grade

This exercise allows you to practice the calculation of the terms of a geometric sequence from its common ratio and its first term.

Exercise example N°1628 :

Let (`u_(n)`) be an arithmetic sequence of common difference 6, and of first term `u_(0)= 1`. Let S be the sum of `u_(3)` to `u_(25)`. S=`u_(3)`+`u_(4)`+`u_(5)`+`. . .`+`u_(25)`.
1. Compute the number of terms in S.
2. Compute S.

numerical sequences 11th Grade 12th Grade

This exercise allows you to practice calculating the sum of the terms of an arithmetic sequence from its common difference and its first term.

Exercise example N°1629 :

Let S be the sum defined by S = `1`.
1. Compute the number of terms in S.
2. Compute S.

numerical sequences 11th Grade 12th Grade

This exercise allows you to practice calculating the sum of the terms of an arithmetic sequence.

Exercise example N°1630 :

Let (`u_(n)`) be a geometric sequence of common ratio -2, and of first term `u_(0)= -2 `. Let S be the sum of `u_(2)` to `u_(14)`. S=`u_(2)`+`u_(3)`+`u_(4)`+`. . .`+`u_(14)`.
1. Calculate `u_(2)`
2. Calculate `u_(14)`.
3. Deduce S.

numerical sequences 11th Grade 12th Grade

This exercise allows you to practice calculating the sum of the terms of a geometric sequence from its common ratio and its first term.

Exercise example N°1631 :

1. Expand and reduce the following polynomial:`(-6+x^2)*(-5-4*x)`.
2. What is its degree ?

polynomial functions algebraic calculus 11th Grade degree

The purpose of this exercise is to practice developing a polynomial and determining its degree.

Exercise example N°1632 :

1. Expand and reduce the following polynomial:`(7+x)^2-1-2*x+x^2+x^3`.
2. What is its degree ?

polynomial functions algebraic calculus 11th Grade degree

The goal of this exercise is to practice developing a polynomial with remarkable identities and determining its degree.

Exercise example N°1633 :

P is the polynomial defined by P(x) =`-4+8*x+3*x^2-x^3`
1. Compute P(-2)
2. Find the polynomial Q such that for any real x, P(x)=(x+2)Q(x)

polynomial functions factoring algebraic calculus 11th Grade factor

The goal of this exercise of algebraic calculation is to factor a polynomial of degree 3 knowing one of its roots.

Exercise example N°1634 :

Compute the roots of P(x) =`-4+8*x+3*x^2-x^3`.

polynomial functions algebraic calculus 11th Grade 12th Grade equation_solver

The goal of this exercise of algebraic calculation is to determine the values for which a polynomial of degree 3 is equal to 0.

Exercise example N°1701 :

Write in algebraic form the complex number Z = `(-4-5*i)/(2+3*i)`

complex numbers 12th Grade complex_number

The goal of this corrected exercise is to write a complex number in its algebraic form z=a+ib.

Exercise example N°1702 :

Compute the real part of the complex number Z = `(2-4*i)/(1+2*i)`

complex numbers 12th Grade real_part

To succeed in this exercise, you must know how to determine the real part of a complex expression.

Exercise example N°1703 :

Calculate the imaginary part of the complex number Z = `(1-3*i)/(5+i)`

complex numbers 12th Grade imaginary_part

The purpose of this exercise is to determine with the help of calculation, the imaginary part of a complex number.

Exercise example N°1704 :

Compute the conjugate of the complex number Z = `(5-2*i)/(1+i)`

complex numbers 12th Grade complex_conjugate

This exercise allows to implement the techniques of calculation of the conjugate of a complex number.

Exercise example N°1705 :

z = `-3+2i`
z' = `5-4i`
Compute `z*z'`.

complex numbers 12th Grade complex_number

The purpose of this exercise is to find the result of arithmetic operations (sum, difference, product) that involve complex numbers.

Exercise example N°1706 :

Compute the imaginary part of the complex number, Z = `-3+2*i`

complex numbers 12th Grade imaginary_part

The objective of this exercise is to find the imaginary part of a complex number from its algebraic form.

Exercise example N°1707 :

Compute the real part of the complex number, Z = `-5+7*i`

complex numbers 12th Grade real_part

The objective of this exercise is to find the real part of a complex number from its algebraic form.

Exercise example N°1708 :

Represent in the complex plane, the point of affix `4+5i`.

complex numbers 12th Grade

The purpose of this graphing exercise is to place in the plane the affix of a complex number.

Exercise example N°1709 :

Express ln(25) as a function of ln(5) .

neperian logarithm functions 12th Grade

The purpose of this corrected exercise is to simplify a neperian logarithm containing a power.

Exercise example N°1710 :

Express `ln(1/27)` as a function of ln(3)

neperian logarithm functions 12th Grade

The goal of this corrected exercise is to simplify a neperian logarithm containing a quotient.

Exercise example N°1711 :

Express `-3/8*ln(1/(27))` as a function of ln(3)

neperian logarithm functions 12th Grade

The goal of this corrected exercise is to simplify the product of a fraction and a neperian logarithm containing a quotient.

Exercise example N°1712 :

Express `-5/8*ln(sqrt(2))` as a function of ln(2)

neperian logarithm functions 12th Grade

The aim of this corrected exercise is to simplify the neperian logarithm of a square root.

Exercise example N°1713 :

Compute an antiderivative of the function `f(x)=7/(9+7*x)` on `RR^+` .

neperian logarithm antiderivatives functions 12th Grade antiderivative

The aim of this corrected exercise is to use the neperian logarithm to calculate one of the primitives of a rational fraction of the first degree.

Exercise example N°1714 :

Compute an antiderivative of the function `f(x)=(8*x)/(1+4*x^2)` on `RR^+` .

neperian logarithm antiderivatives functions 12th Grade antiderivative

The goal of this corrected exercise is to use the neperian logarithm for antiderivative calculation of a rational fraction of degree 2.

Exercise example N°1715 :

Calculate the derivative of the function `ln(x)^5`.

neperian logarithm derivatives of functions functions 12th Grade derivative

The goal of this corrected exercise is to use the neperian logarithm to calculate the derivative.

Exercise example N°1716 :

Calculate the derivative of the function `ln(9+9*x^2)`.

neperian logarithm derivatives of functions functions 12th Grade derivative

The goal of this corrected exercise is to use the neperian logarithm to calculate the derivative.

Exercise example N°1717 :

Simplify the following expression `e^ln(3)+e^ln(4)`.

exponential functions 12th Grade calculator

The goal of this corrected exercise is to use the properties of the exponential and the neperian logarithm to simplify an algebraic expression.

Exercise example N°1718 :

Simplify the following expression `e^ln(8)/e^ln(4)`.

exponential functions 12th Grade calculator

The goal of this corrected exercise is to use the properties of the exponential and the neperian logarithm to simplify an algebraic expression.

Exercise example N°1719 :

Simplify the following expression `e^(ln(8)*ln(4))`.

exponential functions 12th Grade

The goal of this corrected exercise is to use the properties of the exponential and the neperian logarithm to simplify an algebraic expression.

Exercise example N°1731 :

Calculate the derivative of the function `e^(3+5*x^2)`.

exponential derivatives of functions functions 12th Grade derivative

The aim of this corrected exercise is to use the exponential for the calculation of derivatives.

Exercise example N°1740 :

Let f be the function defined by f(x)= `3-2*x^2+x^3` ,compute an antiderivative of f, `F(x)`, with F(x)=0.

antiderivatives functions 12th Grade integral

The aim of this corrected exercise is to use integration methods to calculate one of the antiderivative of a polynomial function.

Exercise example N°2413 :

Determine the reduced equation of the line passing through the points A(3;5 )and B(2;4).

linear functions and affine functions equations of lines and linear systems equations 9th Grade 10th Grade equation_straight_line

The purpose of this exercise is to find the equation of a line from two points.

Exercise example N°3441 :

Let(O,`vec(i)`,`vec(j)`) be a reference frame of the plane. Let A and D be two points of coordinates `(13,8)` and `(7,6)` respectively in this frame, compute the coordinates of the vector `vec(AD)`.

vectors geometry 9th Grade 10th Grade vector_coordinates

The purpose of this corrected exercise is to calculate the coordinates of a vector from the coordinates of two points.

Exercise example N°3442 :

The plane is given an orthonormal reference frame (O,`vec(i)`,`vec(j)`). Let A and D be two points of coordinates (`13`,`8`) and (`7`,`6`) respectively in this frame, calculate the distance between A and D .

vectors geometry 9th Grade 10th Grade vector_norm

The purpose of this corrected exercise is to calculate the distance between two points from their coordinates.

Exercise example N°3443 :

Let(O,`vec(i)`,`vec(j)`) be a reference frame of the plane. Let D and H be two points of coordinates `(2,8)` and `(2,7)` respectively in this frame, compute the coordinates of the middle of the segment [DH].

vectors geometry 9th Grade 10th Grade

The purpose of this corrected analytical geometry exercise is to calculate the coordinates of the midpoint of a segment from coordinates.

Exercise example N°4401 :

In a library open from Tuesday to Saturday inclusive, we counted, day by day, the number of books lent during a week and we obtained the results in the following table:

statistics 9th Grade 10th Grade average

The purpose of this corrected statistics exercise is to practice calculating an arithmetic mean.

Exercise example N°4402 :

After a test, the marks of 26 students have been grouped in the table below :

statistics 9th Grade 10th Grade

The purpose of this corrected statistics exercise is to practice calculating the frequency of a series.

Exercise example N°4403 :

Here are the ages of the employees of a company, give the frequency of employees who are between 25 and 29 years old.

statistics 9th Grade 10th Grade

The purpose of this corrected statistics exercise is to determine from a table the frequency of a series.

Exercise example N°24121 :

Let f be a function whose representation is given opposite. What is the image of -3 by f?

linear functions and affine functions 9th Grade 10th Grade

The purpose of this exercise is to read on a graph the image of a number by a function.

Exercise example N°24122 :

Let f be a function whose representation is given opposite. What is the antecedent by f of -2 ?

linear functions and affine functions 9th Grade 10th Grade

The purpose of this exercise is to read on a graph the antecedent by a function of a number.