Question

1. a table of selected values is given for a one - to - one function ( g(x) ). what is ( g^{-1}(x) )?

\begin{tabular}{|c|c|c|c|c|c|c|}hline( x ) & ( - 4 ) & ( - 2 ) & ( 0 ) & ( 1 ) & ( 5 ) & ( 8 )\hline( g(x) ) & ( 10 ) & ( 8 ) & ( - 3 ) & ( - 1 ) & ( - 4 ) & ( 1 )\hlineend{tabular}

Explanation:

Step1: Recall inverse function definition

For a one - to - one function \(y = g(x)\), the inverse function \(g^{-1}(y)=x\) when \(y = g(x)\). So we need to swap the \(x\) and \(y\) (or \(g(x)\)) values.

Step2: Create the inverse function table

We have the table for \(g(x)\) with \(x\) values \(- 4,-2,0,1,5,8\) and corresponding \(g(x)\) values \(10,8, - 3,-1,-4,1\).
To find \(g^{-1}(x)\), we take the \(g(x)\) values as the new \(x\) values and the original \(x\) values as the new \(g^{-1}(x)\) values.
So we can create a table for \(g^{-1}(x)\):

\(x\) (input for \(g^{-1}\))	\(10\)	\(8\)	\(-3\)	\(-1\)	\(-4\)	\(1\)

Answer:

The inverse function \(g^{-1}(x)\) has the following mapping: when \(x = 10\), \(g^{-1}(10)=-4\); when \(x = 8\), \(g^{-1}(8)=-2\); when \(x=-3\), \(g^{-1}(-3) = 0\); when \(x=-1\), \(g^{-1}(-1)=1\); when \(x = - 4\), \(g^{-1}(-4)=5\); when \(x = 1\), \(g^{-1}(1)=8\). In table form (or as a set of ordered pairs \((10, - 4),(8,-2),(-3,0),(-1,1),(-4,5),(1,8)\))