Download Domain and Restricted Domain SM1 Section 4

Domain and Restricted Domain
SM1 Section 4-6
Date____________________
Objective: Students will graph functions given restricted domains.
Students will identify restricted domains of functions.
The domain of a function is the set of values used to replace the independent variable (x).
The range is the set of values of f(x). Sometimes the domain is restricted (limited to only some
real numbers) and specifically stated. For example each of the following have a restricted
domain specified:
f ( x)  x  3, for x  2, 4, 6
f ( x)  3x ^ 2 for x  0
Sometimes the domain of a function is implied. This means that the domain of the
function is not specified but instead is considered to be all real numbers that make the function
defined in the set of real numbers. For example: the implied domain for the function
f ( x)  x is all positive real numbers and 0,
0, , because these are the only values where
x is possible. Negative real numbers cannot be
part of the domain because the square root of a negative real number is not defined in the real
number system.
The domain of a function can also be determined by looking at a graph: The following are
examples of a graph of a function and its domain. If a graph of a function has arrows on the end
of it, this indicates that the graph continues in that direction. In these graphs, if the graph of the
function reaches the edge of the grid shown it indicates that the function continues in that
direction.
Domain:  ,  
 ,  
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 ,  
Domain:
1, 5
5, 3
 1,  
Practice:
For each of the following identify the domain indicated in the graph. Write the domain
using interval notation.
1.
2.
Domain:_______________
3.
Domain:_______________
Domain:_______________
When given a restricted domain, we can graph any function. The following are examples
of the function, its restricted domain, and its graph.
f ( x)  2x  3 for x  2
f ( x)  2 x  3 for x  0, 2, 4, 6
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Practice:
For each of the following, graph the function using the restricted domain as specified for each.
4.
f ( x)   x  4 for x  1 i.e.
 1,  
5.
f ( x)  2 x for x  0,1, 2
6.
1
f ( x)   x  1 for x  4,  2, 0, 2, 4
2
7.
f ( x)  3x for x  1 i.e.
 ,1
In many real life situations the domain is implied based on the possible values of the
independent variables that make sense in the specific situation.
For example, each of the following indicate the situation that can be modeled with a
function and the implied domain.
-
Eugene works part-time as delivery boy for a law firm. He is only allowed to work up to
20 hours a week and is paid $9.00 per hour. The function that represents his weekly
gross pay is modeled by f ( x)  9 x , where x represents the amount of time he works
during the week.
The implied domain is 0  x  20 , or in interval notation 0, 20 .
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Nadia has a figurine collection. She will display the collection on shelves and has
determined that she needs 5 inches of shelf space for each figurine. She has measured the
walls in her room and has found that she can put up a maximum of 125 inches of shelving
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for the figurines. She has 3 favorite figurines that she knows she will always display.
The function that models the amount of shelving she will need is f ( x)  5 x , where x
represents the number of figurines she will display.
The implied domain is the set of all integers between 3 and 25 inclusive,
i.e. x = 3, 4, 5, …24, 25.
Practice:
For each of the following situations determine the implied domain for the function.
8.
Elise wants to sell scarves in her small business. She will base the price of each scarf on
material cost and labor cost. The price of the materials is the same for all scarves, $3 each. The
length of the scarf can be any length from 30” to 60” and she will charge $0.25 per inch. The
price of each scarf can then be modeled by the function f ( x)  0.25 x  3 , where x represents the
length of the scarf in inches. What is the implied domain of this function?
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9.
Clark has a lawn mowing business. He charges $0.02 per sq foot and an additional $10
cleanup fee. His price can be modeled by the function, f ( x)  0.02 x  10 , where x represents
the number of square feet.
What is the implied domain of this function?
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