The functions f and g are defined by the following tables. Use the tables to evaluate the given composite function.
[tex]f^{-1}(g(7))[/tex]
| x | f(x) |
|---|---|
| -4 | 1 |
| 0 | 4 |
| 1 | 5 |
| 6 | -1 |
| x | g(x) |
|---|---|
| -5 | -5 |
| 1 | -2 |
| 6 | 2 |
| 7 | -1 |
[tex]f^{-1}(g(7))=\square[/tex]
Answer (2)
Find g ( 7 ) from the table: g ( 7 ) = − 1 .
Find f − 1 ( − 1 ) by finding the x value such that f ( x ) = − 1 .
From the table, f ( 6 ) = − 1 , so f − 1 ( − 1 ) = 6 .
Therefore, f − 1 ( g ( 7 )) = 6 .
Explanation
Understanding the Problem We are given two functions, f ( x ) and g ( x ) , defined by tables, and we need to find f − 1 ( g ( 7 )) . This means we first need to find the value of g ( 7 ) , and then find the value x such that f ( x ) is equal to g ( 7 ) .
Finding g(7) First, we look at the table for g ( x ) to find the value of g ( 7 ) . From the table, we see that when x = 7 , g ( x ) = − 1 . So, g ( 7 ) = − 1 .
Finding f^{-1}(g(7)) Now we need to find f − 1 ( g ( 7 )) , which is f − 1 ( − 1 ) . This means we need to find the value of x such that f ( x ) = − 1 . Looking at the table for f ( x ) , we see that when x = 6 , f ( x ) = − 1 . Therefore, f − 1 ( − 1 ) = 6 .
Final Answer Thus, f − 1 ( g ( 7 )) = 6 .
Examples
Composite functions are used in many real-world applications. For example, in manufacturing, a function might convert raw materials into parts, and another function might assemble those parts into a final product. The composite function would then represent the entire manufacturing process from raw materials to finished product. Similarly, in computer graphics, transformations like scaling, rotation, and translation can be represented as functions, and applying these transformations in sequence is equivalent to composing the functions.
To find f − 1 ( g ( 7 )) , we first evaluate g ( 7 ) from its table, resulting in g ( 7 ) = − 1 . Then we find f − 1 ( − 1 ) from the table of f ( x ) , which gives us 6 . Thus, f − 1 ( g ( 7 )) = 6 .
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