Identify the inverse [tex]g(x)[/tex] of the given relation [tex]f(x)[/tex].
[tex]
\begin{aligned}
f(x)= & \{(8,3),(4,1),(0,-1),(-4,-3)\} \
& g(x)=\{(-4,-3),(0,-1),(4,1),(8,3)\} \
& g(x)=\{(-8,-3),(-4,1),(0,1),(4,3)\} \
& g(x)=\{(8,-3),(4,-1),(0,1),(-4,3)\} \
& g(x)=\{(3,8),(1,4),(-1,0),(-3,-4)\}
\end{aligned}
[/tex]
Answer (2)
To find the inverse relation, swap the x and y coordinates in each ordered pair of the original relation.
Given f ( x ) = {( 8 , 3 ) , ( 4 , 1 ) , ( 0 , − 1 ) , ( − 4 , − 3 )} , swap the coordinates to find the inverse.
The inverse relation is g ( x ) = {( 3 , 8 ) , ( 1 , 4 ) , ( − 1 , 0 ) , ( − 3 , − 4 )} .
The correct answer is g ( x ) = {( 3 , 8 ) , ( 1 , 4 ) , ( − 1 , 0 ) , ( − 3 , − 4 )} .
Explanation
Understanding the Problem The problem asks us to find the inverse of the relation f ( x ) = {( 8 , 3 ) , ( 4 , 1 ) , ( 0 , − 1 ) , ( − 4 , − 3 )} . The inverse of a relation is found by swapping the x and y coordinates in each ordered pair.
Finding the Inverse To find the inverse relation g ( x ) , we swap the x and y coordinates in each ordered pair of f ( x ) . So, we have:
( 8 , 3 ) → ( 3 , 8 ) ( 4 , 1 ) → ( 1 , 4 ) ( 0 , − 1 ) → ( − 1 , 0 ) ( − 4 , − 3 ) → ( − 3 , − 4 )
Identifying the Correct Option Therefore, the inverse relation is g ( x ) = {( 3 , 8 ) , ( 1 , 4 ) , ( − 1 , 0 ) , ( − 3 , − 4 )} . Comparing this with the given options, we find that it matches the last option.
Final Answer The inverse of the relation f ( x ) = {( 8 , 3 ) , ( 4 , 1 ) , ( 0 , − 1 ) , ( − 4 , − 3 )} is g ( x ) = {( 3 , 8 ) , ( 1 , 4 ) , ( − 1 , 0 ) , ( − 3 , − 4 )} .
Examples
In real life, inverse relations can be used to convert between different units of measurement. For example, if f ( x ) converts Celsius to Fahrenheit, then g ( x ) converts Fahrenheit to Celsius. Understanding inverse relations helps in converting data back and forth between different representations.
To find the inverse of the relation f ( x ) = {( 8 , 3 ) , ( 4 , 1 ) , ( 0 , − 1 ) , ( − 4 , − 3 )} , we swap the coordinates to get g ( x ) = {( 3 , 8 ) , ( 1 , 4 ) , ( − 1 , 0 ) , ( − 3 , − 4 )} . This matches option four from the given choices. Thus, the correct answer is g ( x ) = {( 3 , 8 ) , ( 1 , 4 ) , ( − 1 , 0 ) , ( − 3 , − 4 )} .
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