A function $f: N \rightarrow N$ is defined by $f(x)=x+3$. Is $f$ one-to-one and onto? What is the pre-image of 100?
Answer (1)
Check if the function f ( x ) = x + 3 is one-to-one by verifying that f ( x 1 ) = f ( x 2 ) implies x 1 = x 2 .
Determine if the function is onto by checking if for every y in N , there exists an x in N such that f ( x ) = y .
Find the pre-image of 100 by solving the equation f ( x ) = 100 for x .
Conclude that the function is one-to-one but not onto, and the pre-image of 100 is 97 .
Explanation
Understanding the Problem We are given a function f : N r i g h t a rro wN defined by f ( x ) = x + 3 , where N represents the set of natural numbers. We need to determine if this function is one-to-one (injective) and onto (surjective). Additionally, we need to find the pre-image of 100, which means finding a value x in N such that f ( x ) = 100 .
Checking if f is One-to-One To check if f is one-to-one, we assume that f ( x 1 ) = f ( x 2 ) for some x 1 , x 2 in N . Then, we have
x 1 + 3 = x 2 + 3
Subtracting 3 from both sides, we get
x 1 = x 2
Since f ( x 1 ) = f ( x 2 ) implies x 1 = x 2 , the function f is one-to-one.
Checking if f is Onto To check if f is onto, we need to determine if for every y in N , there exists an x in N such that f ( x ) = y . In other words, we need to solve the equation x + 3 = y for x and see if the solution is a natural number for all y in N .
Solving for x , we get
x = y − 3
Now, let's consider y = 1 . Then, x = 1 − 3 = − 2 . Since − 2 is not a natural number, there is no x in N such that f ( x ) = 1 . Therefore, the function f is not onto.
Finding the Pre-image of 100 To find the pre-image of 100, we need to find x in N such that f ( x ) = 100 . So, we solve the equation
x + 3 = 100
Subtracting 3 from both sides, we get
x = 100 − 3 = 97
Since 97 is a natural number, the pre-image of 100 is 97.
Conclusion In summary, the function f ( x ) = x + 3 is one-to-one but not onto. The pre-image of 100 is 97.
Examples
Consider a scenario where you are assigning tasks to workers, and each task takes 3 hours to complete. The function f ( x ) = x + 3 represents the total time required for a worker to complete x additional tasks. Determining if the function is one-to-one helps ensure that each worker's workload is uniquely defined. Checking if the function is onto helps determine if all possible total times can be achieved. Finding the pre-image of a specific total time helps determine the number of additional tasks needed to reach that total time. This concept is useful in resource allocation and task management.
