Write a function rule for the table:
| x | f(x) |
|---|---|
| 3 | -1 |
| 4 | 0 |
| 5 | 1 |
| 6 | 2 |
A. [tex]f(x)=x-4[/tex]
B. [tex]f(x)=4-x[/tex]
C. [tex]f(x)=x+4[/tex]
Answer (2)
Test f ( x ) = x − 4 : f ( 3 ) = − 1 , f ( 4 ) = 0 , f ( 5 ) = 1 , f ( 6 ) = 2 . This matches the table.
Test f ( x ) = 4 − x : f ( 3 ) = 1 . This does not match the table.
Test f ( x ) = x + 4 : f ( 3 ) = 7 . This does not match the table.
The correct function rule is f ( x ) = x − 4 .
Explanation
Analyze the problem and data We are given a table of x and f ( x ) values, and we need to determine which of the given function rules correctly maps each x value to its corresponding f ( x ) value. The table provides the following data:
x
f ( x )
3
-1
4
0
5
1
6
2
The possible function rules are:
f ( x ) = x − 4
f ( x ) = 4 − x
f ( x ) = x + 4
Test the first function rule Let's test each function rule with the given x values:
1. f ( x ) = x − 4
f ( 3 ) = 3 − 4 = − 1 (Correct)
f ( 4 ) = 4 − 4 = 0 (Correct)
f ( 5 ) = 5 − 4 = 1 (Correct)
f ( 6 ) = 6 − 4 = 2 (Correct)
Since this function rule produces the correct f ( x ) values for all x values in the table, it is the correct function rule.
Test the second function rule 2. f ( x ) = 4 − x
f ( 3 ) = 4 − 3 = 1 (Incorrect, should be -1)
Since this function rule does not produce the correct f ( x ) value for x = 3 , it is not the correct function rule.
Test the third function rule 3. f ( x ) = x + 4
f ( 3 ) = 3 + 4 = 7 (Incorrect, should be -1)
Since this function rule does not produce the correct f ( x ) value for x = 3 , it is not the correct function rule.
Conclusion The correct function rule is f ( x ) = x − 4 because it produces the correct f ( x ) values for all x values in the given table.
Examples
In real life, function rules can be used to model various relationships between quantities. For example, if you are selling lemonade for $1.50 per cup and your cost to make each cup is 0.50 , t h e p ro f i t f u n c t i o n w o u l d b e P(x) = x - 0.5x , w h ere x$ is the number of cups sold. This function rule helps you determine your profit based on the number of cups you sell. Similarly, function rules can be used to model the relationship between time and distance, temperature and time, or any other two related variables.
The correct function rule for the table is f ( x ) = x − 4 , as it matches all the values provided. The other function rules do not produce the correct outputs based on the given x values. Thus, option A is the right choice.
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