The function $h(x)$ is given below.
$h(x)=\{(3,-5),(5,-7),(6,-9),(10,-12),(12,-16)\}$
Which of the following gives $h^{-1}(x)$ ?
A. $\{(3,5),(5,7),(6,9),(10,12),(12,16)\}$
B. $\{(-5,3),(-7,5),(-9,6),(-12,10),(-16,12)\}$
C. $\{(3,-5),(5,-7),(6,-9),(10,-12),(12,-16)\}$
D. $\{(5,3),(7,5),(9,6),(12,10),(16,12)\}$
Answer (2)
The problem provides a function h ( x ) as a set of ordered pairs.
To find the inverse function h − 1 ( x ) , we swap the x and y coordinates in each ordered pair.
Applying this to the given function, we get h − 1 ( x ) = {( − 5 , 3 ) , ( − 7 , 5 ) , ( − 9 , 6 ) , ( − 12 , 10 ) , ( − 16 , 12 )} .
Therefore, the answer is {( − 5 , 3 ) , ( − 7 , 5 ) , ( − 9 , 6 ) , ( − 12 , 10 ) , ( − 16 , 12 )} .
Explanation
Understanding the Problem The function h ( x ) is given as a set of ordered pairs: h ( x ) = {( 3 , − 5 ) , ( 5 , − 7 ) , ( 6 , − 9 ) , ( 10 , − 12 ) , ( 12 , − 16 )} . We need to find the inverse function h − 1 ( x ) . The inverse function is obtained by swapping the x and y coordinates in each ordered pair.
Finding the Inverse To find the inverse function h − 1 ( x ) , we swap the x and y coordinates in each ordered pair of h ( x ) .
Calculating the Inverse Given h ( x ) = {( 3 , − 5 ) , ( 5 , − 7 ) , ( 6 , − 9 ) , ( 10 , − 12 ) , ( 12 , − 16 )} , we obtain h − 1 ( x ) = {( − 5 , 3 ) , ( − 7 , 5 ) , ( − 9 , 6 ) , ( − 12 , 10 ) , ( − 16 , 12 )} .
Final Answer The inverse function is h − 1 ( x ) = {( − 5 , 3 ) , ( − 7 , 5 ) , ( − 9 , 6 ) , ( − 12 , 10 ) , ( − 16 , 12 )} . Comparing this to the given options, we see that the correct answer is {( − 5 , 3 ) , ( − 7 , 5 ) , ( − 9 , 6 ) , ( − 12 , 10 ) , ( − 16 , 12 )} .
Examples
In real life, inverse functions can be used to convert between different units of measurement. For example, if f ( x ) converts Celsius to Fahrenheit, then f − 1 ( x ) converts Fahrenheit to Celsius. Understanding inverse functions helps us reverse processes and solve for the original input given the output.
The inverse function h − 1 ( x ) is obtained by swapping the coordinates of each pair in h ( x ) . This gives us h − 1 ( x ) = {( − 5 , 3 ) , ( − 7 , 5 ) , ( − 9 , 6 ) , ( − 12 , 10 ) , ( − 16 , 12 )} , which corresponds to option B. Thus, the answer is B.
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