The one-to-one functions [tex]g[/tex] and [tex]h[/tex] are defined as follows.
[tex]
\begin{array}{l}
g=\{(-8,7),(-7,-2),(4,-8),(6,1)\} \\
h(x)=4 x-13
\end{array}
[/tex]
Find the following.
[tex]
\begin{array}{l}
g^{-1}(-8)=4 \\
h^{-1}(x)=\frac{x+13}{4} \\
\left(h \circ h^{-1}\right)(-5)=\square \\
\hline
\end{array}
[/tex]
Answer (1)
The composition of a function and its inverse results in the original input.
Therefore, ( h ∘ h − 1 ) ( − 5 ) = − 5 .
The final answer is − 5 .
Explanation
Understanding the Problem We are given two functions: g and h . The function g is defined as a set of ordered pairs, and the function h is defined by the equation h ( x ) = 4 x − 13 . We are asked to find the value of ( h ∘ h − 1 ) ( − 5 ) .
Using the Inverse Function Property Recall that for any one-to-one function h , the composition of h with its inverse h − 1 is the identity function. That is, ( h ∘ h − 1 ) ( x ) = x for all x in the domain of h − 1 , which is the range of h .
Applying the Property Therefore, ( h ∘ h − 1 ) ( − 5 ) = − 5 .
Conclusion The final answer is − 5 .
Examples
Imagine you are using a lock and key. The function h is like locking something with the key, and h − 1 is like unlocking it. If you lock something and then immediately unlock it, you are back to where you started. Similarly, ( h ∘ h − 1 ) ( − 5 ) means you apply the inverse function to − 5 and then apply the original function to the result, which brings you back to − 5 . This concept is useful in cryptography, where encoding and decoding messages rely on inverse functions.
