A piecewise function [tex]f(x)[/tex] is defined as shown.
[tex]f(x)=\left\{\begin{array}{ll}
-\frac{5}{4} x+90, & 0 \leq x\ \textless \ 40 \\
-\frac{3}{8} x+75, & 40 \leq x \leq 200
\end{array}\right.[/tex]
Which table could be used to graph a piece of the function?
A.
| x | y |
| --- | --- |
| 0 | 90 |
| 16 | 85 |
| 40 | 75 |
B.
| x | y |
| --- | --- |
| 0 | 90 |
| 40 | 40 |
| 200 | 0 |
C.
| x | y |
| --- | --- |
| 40 | 75 |
| 120 | 30 |
| 200 | 0 |
D.
| x | y |
| --- | --- |
| 40 | 60 |
| 160 | 15 |
| 200 | 0 |
Answer (2)
Analyze each table by plugging in the x values into the corresponding part of the piecewise function.
For the first table, f ( 16 ) = 70 , but the table gives 85 , so it's incorrect.
For the second table, f ( 40 ) = 60 , but the table gives 40 , so it's incorrect.
For the third table, f ( 40 ) = 60 , but the table gives 75 , so it's incorrect.
The fourth table matches the piecewise function for all given x values, so the final answer is the fourth table.
Therefore, the answer is the fourth table:
x
y
40
60
160
15
200
0
$
\boxed{
x y 40 60 160 15 200 0 }
$
Explanation
Understanding the Problem We are given a piecewise function and four tables of x and y values. Our goal is to determine which table contains points that lie on the graph of the piecewise function. The piecewise function is defined as:
f ( x ) = { − 4 5 x + 90 , − 8 3 x + 75 , 0 ≤ x < 40 40 ≤ x ≤ 200
We need to check each table to see if the y values correspond to the x values according to the piecewise function's definition.
Checking Each Table Let's analyze the first table:
x
y
0
90
16
85
40
75
For x = 0 , f ( 0 ) = − 4 5 ( 0 ) + 90 = 90 . This matches the table. For x = 16 , f ( 16 ) = − 4 5 ( 16 ) + 90 = − 20 + 90 = 70 . The table has y = 85 , so this table is incorrect.
Let's analyze the second table:
x
y
0
90
40
40
200
0
For x = 0 , f ( 0 ) = − 4 5 ( 0 ) + 90 = 90 . This matches the table. For x = 40 , f ( 40 ) = − 8 3 ( 40 ) + 75 = − 15 + 75 = 60 . The table has y = 40 , so this table is incorrect.
Let's analyze the third table:
x
y
40
75
120
30
200
0
For x = 40 , f ( 40 ) = − 8 3 ( 40 ) + 75 = − 15 + 75 = 60 . The table has y = 75 , so this table is incorrect.
Let's analyze the fourth table:
x
y
40
60
160
15
200
0
For x = 40 , f ( 40 ) = − 8 3 ( 40 ) + 75 = − 15 + 75 = 60 . This matches the table. For x = 160 , f ( 160 ) = − 8 3 ( 160 ) + 75 = − 60 + 75 = 15 . This matches the table. For x = 200 , f ( 200 ) = − 8 3 ( 200 ) + 75 = − 75 + 75 = 0 . This matches the table.
Therefore, the fourth table is the correct one.
Final Answer After carefully evaluating each table against the given piecewise function, we find that the fourth table is the only one where all the points satisfy the function's conditions. Therefore, the correct table is:
x
y
40
60
160
15
200
0
Examples
Piecewise functions are used in real life to model situations where different rules or conditions apply over different intervals. For example, cell phone plans often have different rates for data usage depending on whether you are below or above a certain data limit. Similarly, income tax brackets are a form of piecewise function, where different tax rates apply to different income ranges. Understanding piecewise functions helps in analyzing and predicting outcomes in these scenarios.
After evaluating all the tables against the piecewise function, Table D is the only one where all the points satisfy the function's conditions. Therefore, the correct table is Table D. This table shows the appropriate y-values for the given x-values accepted by the piecewise function.
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