**CHAPTER 1 FUNCTIONS: CHARACTERISTICS AND PROPERTIES**

1.1-Functions

Key Points

Function- A function is a relation in which there is a unique output. This means that each value of the independent variable (domain) must correspond to one, and only one, value of the dependant variable (range)

They can be represented graphically, numerically, or algebraically:

**1.2-Exploring Absolute Value**

1.3 – Properties of Graphs of Functions

Main idea

Compare and contrast properties of various types of functions.

Key Ideas

Functions can be categorized based on their characteristics:

• Domain and range

• X-intercepts and Y-intercepts

• Continuity and discontinuity

• Intervals of increase and decrease

• Symmetry (even/odd)

• End Behaviours

In chapter 1 you will review your knowledge of the properties and characteristics of functions along with their inverses. You will also review your knowledge of graphing functions using transformations. Additionally, you will learn what piecewise functions are and investigate their properties.

This function f(x)=(x+1)^2 -3,i s represented , in this graph

Graph example of a function

Numerical Examples of a Function

(1)Set of Ordered Pairs- {(1,3), (3,5), (-2,9), (5,11)}

(2)Table of Values

(3)Mapping Diagram

Algebraic Example of a Function

y=2sin(3x)+4 or as f(x)=2sin(3x)+4

What you need to know!

• F(x)=y , due to f(x) representing the values of the dependant variable in a function

• You can use (VLT) Vertical Line Test to check if the graph represents a function (A graph represents a function if every vertical line intersects the graph in, at most, one point in the graph)

• Domain and Range of a function depends on the type of function you’re working with

• The Domain and Range of a function are used to show any restrictions if needed in the specific type of function

Key Points

-f(x)= is the Absolute Value function

-On a number line, this function describes distance, f(x), of any number “x” from the origin

Diagram 1: Number line of Absolute Value

Need to Know This

For function f(x)=|x|

-There’s no zero’s located at origin

-The graph is described of two linear functions and as follows

-The graph is symmetric on the y-axis

End Behaviours

-as “x” approaches large positive values, “y” approaches large positive values

-as “x” approaches large negative values, “y” approaches large positive values

-Domain{XER}

-Range{YER |⃒y 0}

-every input absolute value returns an output that is non-negative (|-2|=2)

-This graph is represented by f(x)=|x|.

-f(x)=x, x<0 left side

-f(x)=-x, x≥0 right side

1.4 – Sketching Graphs of Functions

Main idea

Apply transformations to parent functions, and use the best methods to sketch the functions on a graph

Definitions

Turning Point - A point on a curve where the function changes from increasing to decreasing, or vice versa.

Key Ideas

Transformations on a function y = a f(k(x-d)) + c must be performed in this order: horizontal and vertical stretches/compressions/reflections should be performed before translations. All points on the graph of the parent function y = f(x) is changed as follows:

(x, y) ( x/k + d, ay + c)

When using transformations to graph, you can apply both a and k together, and then the translations together, to get the graph in the least possible steps

Example # 1: Sketch y=2(3(x+4)) ^2- 2 step by step.

Example # 2:

The point (1,1) is on the graph of y=x^2 . Find the corresponding coordinates of this point on the graph of y=2(3(x+4)) ^2- 2.

HINT: Whatever is inside the brackets is always opposite and affects the x- coordinate and whatever is outside the brackets affect the y-coordinate.

(x, y ) (1/3x-4, 2y-2)

(1, 1 ) (1/3(1)-4, 2(1)-2)

(1/3(1)-4, 2(1)-2)

(1/3-4, 2-2)

(1/3-4, 2-2)

(3.667, 0)

1.5 Inverse Relations

Tip:

When an inverse relation is also a function, the notation f-1(x) can be used to define the inverse function.

Key Ideas

The inverse function of f (x) is denoted by f-1(x). Function notation can only be used when the inverse is a function.

The graph of the inverse function is a reflection in the y = x.

Need to know

Not all inverse relations are functions. The domain and/or range of the original function may need to be restricted to ensure that the inverse of a function is also a function.

To find the inverse algebraically, write the function equation using y instead of f(x). Interchange x and y.

If (a,b) represents a point on the point on the graph of f(x), then (b,a) represents a point on the graph of the corresponding f-1.

Given a table of values or a graph of a function, the independent and dependent variables can be interchanged to get a table of values or a graph of the inverse relation.

The domain of a function is the range of its inverse. The range of a function is the domain of its inverse.

Remember

The inverse of a function is not always a function. To make the inverse a function, you may have to restrict the domain or range of the original function. If the inverse is already a function, it is referred to as an ``inverse function``. If the inverse is not a function, it is referred to as an ``inverse relation``.

1.6 Piecewise Functions

Definitions

A piecewise function is a function defined by using two or more rules on two or more intervals; as a result, the graph is made up of two or more pieces of similar different functions.

Key Ideas

Some functions are represented by two or more pieces. These functions are called piecewise functions

Each piece of piece wise function is defined for a specific interval in the domain of the function

Need to Know

To graph a piecewise function, graph each piece of the function over the given interval.

A piecewise function can be either continuous or not. If all the pieces of the function join together at the endpoints of the given intervals, then the function is continuous. Otherwise, it is discontinuous at these values of the domain.

Example # 1: Graph the following

piece wise function

1.7 – Exploring Operations with Functions

Main idea

If two functions have domains that overlap each other, they can be added, subtracted, or multiplied in order to produce a new function on the shared domain from these two overlapped functions. This can be done graphically and also algebraically.

Example #1: To add, subtract, or multiply two functions graphically, you have to add, subtract, or multiply the values of the dependent variable/y-coordinate for the identical values of the independent variable/x-coordinate. You may want to use a table of values for this.

1. Use this graph for the following questions.

a) Use the graphs of f and g to sketch the graphs of f + g.

b) Use the graphs of f and g to sketch the graphs of f-g.

c) Use the graphs of f and g to sketch the graphs of fg.

Example #2: To add, subtract, or multiply two functions algebraically, you have to add, subtract, or multiply the expressions from the dependent variable/y-coordinate and then simplify.

Let f = {(-5, 1), (-3, 0), (0, -2), (1, -4), (6, 3)} and let g= {(-3, -1), (-2, 5), (0, 1), (2, 5), (6, 7)}

a) Determine f + g

= {(-3, 0+ (-1)), (0, -2+1), (6, 3+7)}

= {(-3, -1), (0, -1), (6, 10)}

b) Determine f – g

= {(-3, 0-(-1)), (0, -2-1), (6, 3-7)}

= {(-3, 1), (0, -3), (6, -4)}

c) Determine f(g)

= {(-3, 0(-1)), (0, -2(1)), (6, 3(7))}

= {(-3, 0), (0, -2), (6, 21)}

Thanks for attention!

THE END