Topic 1Domain and Range

Unit 6 Topic 1

Explore: Investigating Domain and RangeFor each graph shown describe

i) the x-valuesii) the y-values

Graph 1

Graph 1Sketch the relation onto the x-axis as a straight line to determine the x-values.

i) the x-values include from -3 to 4

Notice that it is closed points at the ends, indicating that the function stops.

Graph 1Sketch the relation onto the y-axis as a straight line to determine the y-values.

ii) the y-values include from -5 to 1

Notice that it is closed points at the ends, indicating that the function stops.

Graph 2

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Graph 2Sketch the relation onto the x-axis as a straight line to determine the x-values.

i) the x-values include all values from negative infinity to positive infinity, all real numbers

Notice that the function has arrows, indicating that the function continues.

Graph 2Sketch the relation onto the y-axis as a straight line to determine the y-values.

ii) the y-values include all values from negative infinity to positive infinity, all real numbers

Notice that the function has arrows, indicating that the function continues.

Graph 3

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Graph 3Sketch the relation onto the x-axis as a straight line to determine the x-values.

i) the x-values include all values from negative infinity to positive infinity, all real numbers

Notice that the function has arrows, indicating that the function continues.

Graph 3Sketch the relation onto the y-axis as a straight line to determine the y-values.

ii) the y-values include all values from negative infinity to 3

Notice that the function has arrows going downward but not at the top, indicating that the function continues in one direction.

Graph 4

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Graph 4Sketch the relation onto the x-axis as a straight line to determine the x-values.

i) the x-values include all values from negative infinity to positive infinity, all real numbers

Notice that the function has arrows, indicating that the function continues.

Graph 4Sketch the relation onto the y-axis as a straight line to determine the y-values.

ii) the y-values include all values from -4 to positive infinity

Notice that the function has arrows going upward but not at the bottom, indicating that the function continues in one direction.

InformationThe domain of a relation is the set of all possible values for the independent variable. The range of a relation is the set of all possible values for the dependent variable. • Symbol Notation

Symbol notation is a formal mathematical way to give the values of the domain and range.

Example 1Stating the Domain and Range From a Graph

For each graph, give the domain and range in symbol notation.a)

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Helpful Hint A solid circle on a graph indicates the point is included.  

Example 1a: Solution

0 3x 0 4y

Example 1Stating the Domain and Range From a Graph

b)

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Helpful Hints For continuous data, use inequality

symbols to indicate the domain and range.

An arrow on a graph indicates that the graph goes on forever in the direction of the arrow.

Example 1b: Solution

x 3y

Example 1Stating the Domain and Range From a Graph

c)

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Example 1c: Solution

x 2y

Example 1Stating the Domain and Range From a Graph

d)

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Example 1d: Solution

0 4x 0 4y

More InformationFunctions can be written using function notation. Function notation is used to distinguish between functions and to evaluate functions. The function may be written in function notation as , which is read “the value of f at x is ”, or f of x is ”. The name of the function is f, with the variable x.

Function notation highlights the relationship between the input and output values. The function takes any input value for x, squares it, multiplies by 2 and adds 1 to produce the output value. For example, f(5) takes the input value 5 and produces the output value or 51. This is written as f(5) = 51.

22 1y x 2( ) 2 1f x x

22 1x 22 1x

2( ) 2 1f x x

22 5 1

Example 2Working With Function Notation

The equations of two functions are given below.

The functions may be expressed in function notation as follows.

2 22 3 5y x y x x

2 2( ) 2 ( ) 3 5f x x g x x x

Example 2Working With Function Notation

Evaluate each of the following.

a)

b)

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( 3)

(5)

f

g

Example 2: Solutiona)

b)

2

2

2

2

( ) 3 5

(5) 5 3 5 5

(

( ) 2

( 3) 2 3

( 3) 18

5) 15

g x

f x x

f

f

x x

g

g

Example 3Stating the Domain and Range in a Real-Life Situation

a)The graph of the relation is shown below.

For the graph shown above, state the following in symbol notation. i. domain

ii. range

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2( ) 4.9 12f x x

i. domain

ii. range

Example 3a: Solution

x

12y

Example 3Stating the Domain and Range in a Real-Life Situation

b) In Southern Alberta, near Fort MacLeod, you will find the famous Head-Smashed-In Buffalo Jump. In a form of hunting, Blackfoot once herded buffalo and then stampeded the buffalo over the cliffs. If the height of a buffalo above the base of the cliff, h, in metres, can be modelled by the function

where t is the time in seconds after the buffalo jumped, how long was the buffalo in the air, to the nearest hundredth of a second?

2( ) 4.9 12h t t

Example 3Stating the Domain and Range in a Real-Life Situation

b) Only a portion of the graph above is appropriate for this situation. In this context, state the restriction in symbol and interval notation.

i. domain

ii. range

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Helpful Hint In a real-life situation the domain and range restrictions are set by the context of the situation.

i. domain

ii. range

Example 3b: Solution

0 1.5x

0 12y

Helpful Hint In a real-life situation the domain and range restrictions are set by the context of the situation.

Time (seconds)

Height (metres)

Example 3Stating the Domain and Range in a Real-Life Situation

c) Explain why the domain and range are different in parts a) and b).

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Example 3c: SolutionStating the Domain and Range in a Real-Life Situation

c) Explain why the domain and range are different in parts a) and b).

Part a) is the graph of the function with no restrictions.

Part b) is for the given context and is restricted:• time (x-values) are only positive numbers starting at 0

seconds,• height (y-values) start at the top of the cliff 12 m and end at

ground level 0 m.

Need to Know:• The domain is the set of all possible values for the

independent variable, x, in a relation.• The range is the set of all possible values for the

dependent variable, y, in a relation.• The domain and range can be expressed in symbol

notation.

Need to Know:• Each function can be expressed in function notation.

For example, the function can be expressed in function notation as . The function f takes an input value, x, multiplies the squared value by 2 and adds 5 to produce the output value.

• For , to evaluate , replace the input value, x, by 4 to obtain the output value .

You’re ready! Try the homework from this section.

22 5y x 2( ) 2 5f x x

2( ) 2 5f x x (4)f

2(4) 2 4 5 32 5 37f