Question

below is the table for the function $f(x)$.

$x$	3	4	8	10	14
$y$	1	6	9	13	16

choose the one table below which is the inverse function $f^{-1}(x)$.

• option 1:

$x$	$1/1$	$1/6$	$1/9$	$1/13$	$1/16$
$y$	1	6	9	13	16

• option 2:

$x$	14	10	8	4	3
$y$	16	13	9	6	1

• option 3:

$x$	3	4	8	10	14
$y$	$1/1$	$1/6$	$1/9$	$1/13$	$1/16$

• option 4:

$x$	1	6	9	13	16
$y$	3	4	8	10	14

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Explanation:

Step1: Recall inverse function table rule

To find the inverse function \( f^{-1}(x) \) from a table of \( f(x) \), we swap the \( x \)-values (input) and \( y \)-values (output) of \( f(x) \).

For the original function \( f(x) \), the table is:

\( x \)	3	4	8	10	14
\( y \)	1	6	9	13	16

Step2: Swap \( x \) and \( y \) values

So for \( f^{-1}(x) \), the \( x \)-values should be the \( y \)-values of \( f(x) \) and the \( y \)-values should be the \( x \)-values of \( f(x) \).

Let's check each option:

• First option: Incorrect, as it involves reciprocals which is not related to inverse function of a table (swapping is needed, not reciprocal).
• Second option: Incorrect, as it reverses the order of \( x \)-values of \( f(x) \) but does not swap \( x \) and \( y \).
• Third option: Incorrect, as it involves reciprocals which is wrong.
• Fourth option:

\( x \)	1	6	9	13	16
\( y \)	3	4	8	10	14

This swaps the \( x \) and \( y \) values of the original function's table, so it represents \( f^{-1}(x) \).

Answer:

The table for \( f^{-1}(x) \) is the fourth option:

\( x \)	1	6	9	13	16
\( y \)	3	4	8	10	14	(the last table option)