Types Of Functions

Given, Here, we see that for every odd values of , it will give zero. It means that it is a many one function. For every even values of , we will get a set of integers . So, it is onto. Hence, it is surjective but not injective.

Step 1: Understanding the Concept:
To determine the nature of the function, we check for injectivity (one-to-one) by seeing if implies . We check for surjectivity (onto) by determining if the range of the function is equal to the codomain ( ).

Step 2: Detailed Explanation:
1. Injectivity: Assume : Thus, the function is Injective. 2. Surjectivity: Let . We solve for in terms of : For to be defined, . Additionally, the domain excludes positive integers. If we pick a value of such that results in a positive integer (e.g., if , then ), that has no pre-image in set . Furthermore, is never reached because the horizontal asymptote is . Since the range does not cover all of , it is not surjective.

Step 3: Final Answer
The function is injective but not surjective.

Step 1: Understanding the Concept:
A function is invertible if and only if it is bijective, meaning it is both injective (one-to-one) and surjective (onto). If a function fails the Horizontal Line Test (meaning multiple values result in the same value), it is not injective and thus not invertible.

Step 2: Detailed Explanation:
1. Option (a): is a linear function. Every gives a unique , and every has a pre-image. It is bijective and invertible. 2. Option (b): with domain . Check injectivity: - and . - Since but , the function is many-to-one. - Because it is not injective, it cannot be bijective, making it not invertible. 3. Option (c): with domain . In this domain, is always positive. The function is strictly decreasing and one-to-one. It is invertible.

Step 3: Final Answer
The function defined on is not invertible.