Manish writes the functions [tex]g(x)=\sqrt[3]{-x}-72[/tex] and [tex]h(x)=-(x+72)^3[/tex]. Which pair of expressions could Manish use to show that [tex]g(x)[/tex] and [tex]h(x)[/tex] are inverse functions?
A. [tex]\sqrt[3]{x^3}-72[/tex] and [tex]-(\sqrt[3]{-x}+72)^3[/tex]
B. [tex]\sqrt[3]{-x^3}-72[/tex] and [tex]-(\sqrt[3]{-x}+72)^3[/tex]
C. [tex]\sqrt[3]{(x+72)^3}-72[/tex] and [tex]-(\sqrt[3]{-x}-72+72)^3[/tex]
D. [tex]\sqrt[2]{-(x+72)^3}-72[/tex] and [tex]-(\sqrt[3]{-x}-72+72)^3[/tex]
Answer (1)
To show that g ( x ) and h ( x ) are inverse functions, we need to verify that g ( h ( x )) = x and h ( g ( x )) = x .
Compute g ( h ( x )) by substituting h ( x ) into g ( x ) : g ( h ( x )) = 3 ( x + 72 ) 3 − 72 .
Compute h ( g ( x )) by substituting g ( x ) into h ( x ) : h ( g ( x )) = − ( 3 − x − 72 + 72 ) 3 = x .
The pair of expressions that could be used to show that g ( x ) and h ( x ) are inverse functions is 3 ( x + 72 ) 3 − 72 and − ( 3 − x − 72 + 72 ) 3 . 3 ( x + 72 ) 3 − 72 and − ( 3 − x − 72 + 72 ) 3
Explanation
Understanding Inverse Functions We are given two functions, g ( x ) = 3 − x − 72 and h ( x ) = − ( x + 72 ) 3 . We want to determine which pair of expressions can be used to show that g ( x ) and h ( x ) are inverse functions. To do this, we need to verify that g ( h ( x )) = x and h ( g ( x )) = x .
Computing g(h(x)) First, let's compute g ( h ( x )) . We substitute h ( x ) into g ( x ) : g ( h ( x )) = g ( − ( x + 72 ) 3 ) = 3 − ( − ( x + 72 ) 3 ) − 72 = 3 ( x + 72 ) 3 − 72.
Computing h(g(x)) Next, let's compute h ( g ( x )) . We substitute g ( x ) into h ( x ) : h ( g ( x )) = h ( 3 − x − 72 ) = − ( 3 − x − 72 + 72 ) 3 = − ( 3 − x ) 3 = − ( − x ) = x .
Matching the Calculations with the Options Now we need to check which of the given options matches our calculations. We have g ( h ( x )) = 3 ( x + 72 ) 3 − 72 and h ( g ( x )) = − ( 3 − x − 72 + 72 ) 3 .
Comparing this to the given options:
Option 1: 3 x 3 − 72 and − ( 3 − x + 72 ) 3 Option 2: 3 − x 3 − 72 and − ( 3 − x + 72 ) 3 Option 3: 3 ( x + 72 ) 3 − 72 and − ( 3 − x − 72 + 72 ) 3 Option 4: 2 − ( x + 72 ) 3 − 72 and − ( 3 − x − 72 + 72 ) 3
Option 3 matches our calculations.
Final Answer Therefore, the correct pair of expressions is 3 ( x + 72 ) 3 − 72 and − ( 3 − x − 72 + 72 ) 3 .
Examples
Inverse functions are useful in cryptography. For example, if a message is encrypted using a function, the inverse function can be used to decrypt the message. In this case, if we encrypt a message x using the function h ( x ) = − ( x + 72 ) 3 , we can decrypt it using the function g ( x ) = 3 − x − 72 . This ensures that the original message can be recovered accurately.
