Answered

Select the correct answer.



Consider the function:




$x$		-1012

$f(x)$		-23813






Which function could be the inverse of function $f$?



A.


$x$		-23813

$q(x)$		-1012






B.


$x$		10-1-2

$s(x)$		-23813






C.


$x$		-1012

$p(x)$		2-3-8-13






D.


$x$		2-3-8-13

$r(x)$		10-1-2

Verified Answer :

  • Extract the coordinate pairs \((-1, -2), (0, 3), (1, 8), (2, 13)\) from the given function table.
  • Identify the inverse function points by swapping the \(x\) and \(y\) values to get \((-2, -1), (3, 0), (8, 1), (13, 2)\).
  • Select the table that contains these pairs as \((x, q(x))\), which is the first option provided.
  • The correct table is \(\boxed{\text{Option 1}}\).

Explanation

  1. Analyzing the Function Data

To find the inverse of the function \(f(x)\), we must identify the mapping that reverses the original input-output relationship of \(f\). Given the function \(f(x)\) defined by the table:
\[ @@TABLE_TOKEN_0@@ \]
The function maps the inputs \(\{-1, 0, 1, 2\}\) to the outputs \(\{-2, 3, 8, 13\}\). We express these as ordered pairs \((x, f(x))\):
\((-1, -2), (0, 3), (1, 8), (2, 13)\).

  1. Determining the Inverse Function Points

The inverse function, denoted by \(f^{-1}(x)\), is defined such that if \(f(a) = b\), then \(f^{-1}(b) = a\). By applying this definition to our set of points \(\{(x, f(x))\}\), we swap the \(x\) and \(y\) coordinates to obtain the set of points for the inverse function:
\((-2, -1), (3, 0), (8, 1), (13, 2)\).
This means that for the inverse function:

  • When \(x = -2\), the function value is \(-1\).
  • When \(x = 3\), the function value is \(0\).
  • When \(x = 8\), the function value is \(1\).
  • When \(x = 13\), the function value is \(2\).
  1. Verifying the Correct Inverse Table

We compare the required inverse points \(\{(-2, -1), (3, 0), (8, 1), (13, 2)\}\) with the given options.
The first option provides the table:
\[ @@TABLE_TOKEN_1@@ \]
This table matches our derived set of points perfectly. Each input \(x\) in the table for \(q(x)\) correctly maps to the value that was the original input for \(f(x)\). Therefore, \(q(x)\) represents the inverse of \(f(x)\). The final answer is the first option.

Examples

Inverse functions are essential in fields where you need to reverse a process. For example, if a function \(f(t)\) represents the temperature of a liquid at time \(t\), the inverse function \(f^{-1}(T)\) allows you to determine at what time the liquid reached a specific temperature \(T\). This is useful in laboratory experiments and industrial manufacturing to monitor and control processes accurately.

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