Question 3
Given the following functions, use function composition to determine if and are inverse
fucntions.

(a)
(b) Details
© Thus Select an answer the inverse function of
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To check if and are inverse functions, we need to compute their compositions.

First, let’s find :

So, .

Next, let’s find :

To check if this simplifies to , notice that we need to analyze . This does not simplify back to , which means that .

Thus, given that and , we can conclude that is not the inverse function of .

Now, for some additional fun facts:

Did you know that the concept of inverse functions can be traced back to ancient civilizations? Mathematicians like Al-Khwarizmi, in the 9th century, laid down the groundwork for solving equations systematically. This exploration of functions blossomed further in the Renaissance when algebra began to develop into a science of its own!

In the real world, understanding inverse functions is crucial in fields like cryptography. Inverse functions help in creating encoding and decoding methods that keep information secure. So when you’re sending a secret message, the cryptographic functions ensuring your message arrives safely are just a modern-day application of those ancient mathematical principles!