Consider the function [tex]$f$[/tex].
[tex]$f(x)=\left\{\begin{array}{ll}\left(\frac{1}{3}\right)^x-1, & x \leq 0 \\ 3^x-2, & x \textgreater 0\end{array}\right.$[/tex]
Complete the table of values for function [tex]$f$[/tex], and then plot the ordered pairs on the graph.
| x | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| f(x) | | | | | |
Answer (2)
Evaluate f ( x ) for x = − 2 , − 1 , 0 using f ( x ) = ( 3 1 ) x − 1 .
Evaluate f ( x ) for x = 1 , 2 using f ( x ) = 3 x − 2 .
Complete the table with the calculated values: f ( − 2 ) = 8 , f ( − 1 ) = 2 , f ( 0 ) = 0 , f ( 1 ) = 1 , f ( 2 ) = 7 .
Plot the points ( − 2 , 8 ) , ( − 1 , 2 ) , ( 0 , 0 ) , ( 1 , 1 ) , ( 2 , 7 ) on the graph.
Explanation
Understanding the Problem We are given a piecewise function f ( x ) and asked to complete a table of values and plot the corresponding points on a graph. The function is defined as:
0 \end{array}\right."> f ( x ) = { ( 3 1 ) x − 1 , 3 x − 2 , x ≤ 0 x > 0
We need to evaluate f ( x ) for x = − 2 , − 1 , 0 , 1 , 2 .
Calculating Function Values For x = − 2 , − 1 , 0 , we use the first part of the piecewise function, f ( x ) = ( 3 1 ) x − 1 .
When x = − 2 :
f ( − 2 ) = ( 3 1 ) − 2 − 1 = 3 2 − 1 = 9 − 1 = 8
When x = − 1 :
f ( − 1 ) = ( 3 1 ) − 1 − 1 = 3 1 − 1 = 3 − 1 = 2
When x = 0 :
f ( 0 ) = ( 3 1 ) 0 − 1 = 1 − 1 = 0
For x = 1 , 2 , we use the second part of the piecewise function, f ( x ) = 3 x − 2 .
When x = 1 :
f ( 1 ) = 3 1 − 2 = 3 − 2 = 1
When x = 2 :
f ( 2 ) = 3 2 − 2 = 9 − 2 = 7
Completing the Table Now we can complete the table:
x
-2
-1
0
1
2
f(x)
8
2
0
1
7
The ordered pairs are ( − 2 , 8 ) , ( − 1 , 2 ) , ( 0 , 0 ) , ( 1 , 1 ) , ( 2 , 7 ) .
Plotting the Points Finally, we plot the points ( − 2 , 8 ) , ( − 1 , 2 ) , ( 0 , 0 ) , ( 1 , 1 ) , ( 2 , 7 ) on the graph.
Examples
Piecewise functions are used in real-world scenarios to model situations where the rule or relationship changes based on the input. For example, cell phone plans often have different rates for data usage based on the amount of data consumed. The cost might be one rate for the first few gigabytes and then a different rate for additional usage. Similarly, income tax brackets are a classic example of a piecewise function, where the tax rate changes as income increases. Understanding piecewise functions helps in analyzing and predicting outcomes in these types of situations.
We evaluated the piecewise function f ( x ) for x = − 2 , − 1 , 0 , 1 , 2 and found the values: f ( − 2 ) = 8 , f ( − 1 ) = 2 , f ( 0 ) = 0 , f ( 1 ) = 1 , and f ( 2 ) = 7 . The completed table contains these results, and the corresponding points can be plotted for visualization. Understanding piecewise functions helps in analyzing diverse scenarios in mathematics.
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