Grades 9/10/11 Introduction to Functions

Function notation. Domain and range. Transformations of functions. Inverses of functions. Inequalities, absolute values, and reciprocals.

This is one of seven strands of the CEMC Grade 9/10/11 courseware. The other strands and more information about this courseware is available on the Grade 9/10/11 homepage.

Unit 1: Representing Functions

In this lesson, the concept of a function is introduced. We look at the definition of a function and different ways to represent functions.

In this lesson, function notation is introduced. We also explore concepts previously learned about linear and quadratic functions, now using function notation.

In this lesson, we will define domain and range. Set notation will be introduced and used to describe the domain and range of various functions, including quadratic functions.

In this lesson, we will continue our study of domain and range. The domain and range of square root functions and rational functions will be investigated.

All of the pencil and paper practice exercises, answers, and solutions for this unit are reproduced here.

This is a collection of additional, and sometimes challenging, problems that extend the material covered in this unit, connect material from different lessons, and further explore real-world applications.

Unit 2: Transforming and Graphing Functions

In this lesson, the graphs of the quadratic function \(f(x)=x^2\), the square root function \(f(x)=\sqrt x\), and the reciprocal function \(f(x)= \dfrac{1}{x}\) will be sketched. The domain and range of each of these functions will also be discussed in relation to the graphs.

In this lesson, we will discuss horizontal and vertical translations and their effect on functions.  We will express translations using function notation and sketch graphs by applying transformations to base functions.

In this lesson, we will discuss reflections in the ‌\(y\)-axis as well as horizontal stretches and compressions, and their effect on functions.  We will express all of these types transformations using function notation and use them to sketch graphs. 

In this lesson, we will discuss reflections in the ‌\(x\)-axis as well as vertical stretches and compressions, and their effect on functions.  We will express all of these types of transformations using function notation and use them to sketch graphs. Comparisons will be made between reflections in the ‌\(x\)-axis and ‌\(y\)-axis, and between vertical and horizontal stretches or compressions. 

In this lesson, we will be discussing all of the types of transformations together. Transformations on a function \(f(x)\) will be identified from the notation \(y=af(b(x-h))+k\) and applied in an appropriate order to sketch graphs.

All of the pencil and paper practice exercises, answers, and solutions for this unit are reproduced here.

This is a collection of additional, and sometimes challenging, problems that extend the material covered in this unit, connect material from different lessons, and further explore real-world applications.

Unit 3: Inverses of Functions

In this lesson, the concept of the inverse of a function is introduced. Inverses are constructed and analyzed for functions described using tables, mapping diagrams, or graphs. We also take a first look at inverses of functions described algebraically.

In this lesson, inverses of linear functions are calculated. The relationship between the domain and range of a function and its inverse is also explored.

In this lesson, we study functions whose inverses are not functions, with a special focus on quadratic functions and their inverses.

All of the pencil and paper practice exercises, answers, and solutions for this unit are reproduced here.

This is a collection of additional, and sometimes challenging, problems that extend the material covered in this unit, connect material from different lessons, and further explore real-world applications.

Unit 4: Inequalities, Absolute Values, and Reciprocals

In this lesson, students will solve both linear and quadratic single variable inequalities. Students will apply multiple strategies to solve quadratic inequalities in different contexts, and they will be introduced to interval notation as a way of expressing their solutions. 

In this lesson, students will interpret and graph linear and quadratic inequalities in two variables. They will use these types of inequalities to solve problems in different contexts.  

In this lesson, the reciprocal function is introduced for linear and quadratic functions. Their graphs are analyzed with a focus on the domain, range, and asymptote(s) of the reciprocal function.

In this lesson, the absolute value function is introduced. Absolute value functions are explored as transformations of the base absolute value function and as piecewise functions. Particular attention is paid to the absolute value of linear and quadratic functions.

In this lesson, absolute value equations are solved, both graphically and algebraically. Single-variable equations that include one absolute value with either a linear or quadratic argument will be considered.

All of the pencil and paper practice exercises, answers, and solutions for this unit are reproduced here.

This is a collection of additional, and sometimes challenging, problems that extend the material covered in this unit, connect material from different lessons, and further explore real-world applications.