Free Online Tutorials on Functions and Algebra

Free analytical Tutorials using step by step approach with examples and matched exercises are presented here. Detailed solutions to the examples are also included. Several tutorials have been designed so that they can be used with the applets tutorials in this site.

Applications, Graphs, Domain and Range of Functions

• Applications of Functions. Several problems and exercises to illustrate possible applications of functions.
• Applications and Use of the Inverse Functions. Examples on how to apply and use inverse functions in real life situations and solve problems in mathematics.
• Graphs of Functions - Tutorials. Identify the graph of an algebraic function given by its formula.
• Find Range of Functions.
• Find Range of Square Root Functions.
• Find Range of Quadratic Functions.
• Domain of a Graph. Examples with detailed solutions included.
• domain and range of basic functions. A table of domain and range of basic and useful functions.
• Domain and Range of Functions and the Answers to matched problems 1,2,3 and 4. Tutorial, with detailed solutions, on how to find the domain and range of real valued functions.
• Find the Domain of logarithmic Functions
• Find the Domain of Rational Functions
• Find the Domain of Square Root Functions
• Evaluate Functions. Tutorial on how to find the value of a function using its formula.
• x and y Intercepts Of Graphs. How to determine the x and the y intercepts of the graphs of functions? Examples with detailed solutions.
• How to Find Zeros of a Function?. How to find the zeros of functions; tutorial with examples and detailed solutions.
• Questions on one to one Functions. Several questions with detailed solutions as well as exercises with answers on how to prove that a given function is a one to one function.

Algebra

• Algebra Tutorial and the detailed solutions to the matched problems.
• Basic Rules and Properties of Algebra. Rules and Properties such as commutativity, associativity and distributivity of algebra with examples.
• Mathematical Induction - Problems With Solutions. Several problems, with detailed solutions, on mathematical induction are presented.
• Completing the Square of Quadratic Expressions. Tutorial on completing the square of quadratic expressions.

Lines

• Slopes of Perpendicular Lines Questions. A detailed tutorial on proof that the slope of two parallel lines are equal and also questions with detailed solution.
• Slopes of Perpendicular Lines Questions. A detailed tutorial on the proof of the fact that two perpendicular lines have slopes such that one is the negative of the reciprocal of the other and also questions with detailed solutions.

Exponential and Logarithmic Functions and Equations

• Solve Logarithmic Equations - Detailed Solutions.. Solve logarithmic equations including some challenging questions. Detailed solutions provided.
• Exponential Functions Questions with Solutions.
• Solve Exponential and Logarithmic Equations, examples with detailed solutions and explanations.
• Tutorial on logarithmic functions.. Tutorial on logarithmic functions with detailed solutions of examples.
• Convert Logarithms and Exponentials.. Tutorial, with examples, on the conversion of logarithmic expression into exponentials and vice versa is.
• Euler Constant e.. Definition of the Euler constant e approximately equal to 2.7182818... .
• natural logarithm
• natural exponential

Quadratic Functions Problems

• Problems, with detailed solutions, involving quadratic functions, parabola, vertex and intercepts.
• Vertex and Intercepts of Quadratic Functions .
• Quadratic Functions Problems with Solutions.

Floor and Ceiling Functions

• Floor Function. The floor function floor(x) is defined as the function that gives the highest integer less than or equal to x.
• Ceiling Function. The ceiling function ceiling(x) is defined as the function that gives the smallest integer greater than or equal to x.

Polynomial Functions

• Polynomial Division. The long division process of Polynomials is presented using an example.
• Synthetic Polynomial Division.
• Applications of Synthetic Polynomial Division. l
• Online Polynomial Division Calculator.
• Introduction to Polynomials. Definition of degree, leading coefficient, monomial, binomial, trinomial and constant, linear, quadratic and cubic polynomial.
• Add, Subtract and simplify Polynomials. A detailed tutorial on what are like terms, how to group them and how to add and subtract polynomials.
• Factor Polynomials. Tutorial, with detailed solutions, on factoring polynomials
• Polynomial Functions, Zeros, Factors and Intercepts. Find polynomial functions given their zeros, graphs and some other information.
• Find Zeros of Polynomial Functions - Problems
• Graphs of Polynomials Functions
• Graphs of Second Degree Polynomials. The graphs of several second degree polynomials are shown along with questions and answers.
• Graphs of Third Degree Polynomials. The graphs of several third degree polynomials are shown along with questions and answers.

Rational Functions

• Find rational functions and answers to the matched exercises.
• Graphs of Rational Functions - Tutorials. Identify graphs of rational functions.

Circle, Ellipse, Parabola and Hyperbola Equations

• Tutorial on Equation of Circle and the answers to the matched exercises: Find equation of a circle, radius and center.
• Tutorial on equation of circle(2) and the answers to the matched exercises: Find equation of the circle.
• Tutorial on equation of circle(3). Find the equation of the circle.
• Find the Points of Intersection of two Circles.This is tutorial on finding the points of intersection of two circles given by their equations; general solution.
• Find the Points of Intersection of a Circle with a Line.
• Find the Points of Intersection of a Circle and an Ellipse.This is tutorial on finding the points of intersection of a circle and an ellipse given by their equations.
• Find the Points of Intersection of Two Ellipses.
• Find Points of Intersection of an Ellipse and a line.
• Ellipse Problems. Problems related to the ellipse equation.
• Parabola Problem with Solution (1). A problem involving a parabola and a line is presented along with detailed solution.
• Find the Points of Intersection of a Parabola with a Line.
• Hyperbola Problems. Problems with detailed solutions on the hyperbola equation.
• Find the Points of Intersection of Two Hyperbolas.
• Points of Intersection of a Hyperbola and a Line.

Find Inverse Functions

• Find Inverse Of Cube Root Functions.
• Find Inverse Of Square Root Functions.
• Find Inverse Of Logarithmic Functions.
• Find Inverse Of Exponential Functions.
• Properties of Inverse Functions. Tutorial on the properties of inverse functions with some examples.
• Find inverse function (1)- Tutorial. Find inverse functions analytically.
• Find inverse function (2)- Tutorial. Find inverse functions analytically.
• Inverse of Quadratic Functions. Examples with detailed solutions on how to find the inverse of quadratic functions with restricted domains.

Composition of Functions

• Composition of Functions. Definition, examples with detailed solutions and properties of composite functions.
• Tutorials Composition of Functions. Find the composition of functions and its domain.
• Applications of Composition of Functions. Examples with solutions
• Composition of Functions Questions with detailed solutions.

Numbers

• Prime Factors. Several examples with detailed solutions on how to factor a positive integer using prime numbers.
• Even and Odd Numbers Problems. Problems with detailed solutions and exercises on even and odd numbers.
• Divisibility Rules. How to use rules and test a whole number for divisibility by 2, 3, 4, 5, 6, 8, 9 and 10.
• Divisibility Rule for 7 Examples and Questions.
• Divisibility Questions With Solutions. Tutorials with SAT maths test like questions and their detailed solutions on divisibility.

Fractions

• Fractions Questions and Problems with Solutions

Percentages

• Percent Maths Problems Several word problems on percentage with detailed solutions are presented.
• Percent Maths Questions Several questions on percentages, with detailed solutions are presented.
• Convert Percentages, Fractions and Decimals Tutorial on how to convert percentages, fractions and decimals. Questions with detailed solutions as well as exercises with answers are presented.

Vectors

• Vector Addition and Scalar Multiplication. A tutorial on how to find components of a vector , add two vectors, multiply a vector by a scalar. Examples, solutions, exercises and answers are included.
• Dot Product of Two Vectors and Applications. A tutorial on the dot product of two vectors and how it is applied to find angle between two vectors.
• Find magnitude and direction of vectors.

Matrices

• Matrix Addition and Multiplication. A tutorial on how to add, multiply by a scalar and multiply matrices with examples, solutions, exercises and answers.
• The Inverse of a Square Matrix. A tutorial on how to find the inverse of any square matrix. The proof of a formula that gives the inverse of an invertible 2 x 2 matrix is also included.

Complex Numbers

• Complex Numbers - Basic Operations. A tutorial on how to find the conjugate of a complex number and add, subtract, multiply, divide complex numbers supported by online calculators.
• Complex Numbers in Exponential Form.

Determinant of a Matrix and Cramer's rule

• Calculate Determinant of a Matrix. A tutorial to calculate the determinant of 2 by 2 and 3 by 3 matrices.
• Cramer's Rule to Solve Systems of Equations. A tutorial on proving Cramer's rule for a 2 by 2 systems of linear equations. Presentation of the rule for a 3 by 3 systems of equations. An example on using Cramer's rule is presented.

Rational Expressions

• Rational Expressions. Define rational expressions and their domains.

Absolute Value

• Simplify Absolute Value Expressions. Tutorial on how to simplify expressions with absolute value.
• Solve Equations, Systems of Equations and Inequalities

Slopes and Lines

• Slope of Lines Questions with Solutions and detailed explanations.
• Equations of Lines in Different Forms . Tutorial on the different forms of the equations of lines.
• Equations of Lines Questions with Solutions .