Problem 36 - Algebra and Trigonometry Real Mathematics, Real People

Question

Does the function have an inverse? Explain. Domain Range \(1 / 2\) hour \(\longrightarrow \$ 40\) 1 hour 2 hours \(\longrightarrow . \$ 70\) 4 hours \(\longrightarrow \$ 120\)

Step-by-Step Solution

Answer

Yes, the function does have an inverse because it is both one-to-one (injective) and onto (surjective)

Step 1: Verify if the function is injective (one-to-one)

To check if the function is injective, we need to make sure that no two different inputs produce the same output. Looking at the given input-output pairs, none of the outputs repeat for different inputs, thus this function is indeed injective.

Step 2: Verify if the function is surjective (onto)

To confirm if the function is surjective, it should cover all the elements of the range. However, in this exercise, a specific range has not been given. In such cases, we can say that the range is just the set of outputs that we have, thus this function will effectively be considered as surjective.

Step 3: Conclusion about the Inverse

Since the function is both injective (one-to-one) and surjective (onto), according to the definition of a bijective function, we can conclude that the function does indeed have an inverse.

Key Concepts

Injective Functions

An injective function, also known as a one-to-one function, is one where each element of the domain maps to a unique element in the range. This means no two different inputs in the domain will map to the same output in the range.

• Example Observation: Consider the mapping from time worked to earnings. If working for 1 hour earns you $50, and working for 2 hours earns you $100, then no two different time intervals should result in the same earnings in a truly injective function.
• Verification Method: Check the provided input-output pairs. If each input is linked to a unique outcome and no output is repeated unless it's from the same input, then the function is injective.

In essence, injective functions ensure that different inputs do not "collide" at the same output. By checking this condition in the provided exercise, we verified that indeed no repetition occurred among varied results, thereby confirming the injectivity of the function.

Surjective Functions

Surjective functions, or onto functions, are those in which every element of the function's range is associated with some element in the domain. In simpler terms, every possible result is covered by some input.

• Example Realization: Picture a scenario where each earning amount on a chart corresponds to at least one period of work duration. If you see an earning like $70 and can identify a corresponding (even theoretical) time worked, the function fulfills the criteria of being surjective.
• Verification Approach: Without a defined full range, surjectivity might be concluded by assuming the present outputs form the whole range, especially in limited context exercises.

In the given task, although the wide range wasn't explicitly defined, considering our observed outputs as the complete range allowed us to determine that each output was potentially reachable by at least one input, characterizing the function as surjective.

Bijective Functions

A bijective function is one that is both injective (one-to-one) and surjective (onto). This creates a perfect "one-to-one correspondence" between the domain and the range. Every element in the domain maps to a unique element in the range, and every element in the range is mapped by some element of the domain.

• Example Understanding: Imagine a scenario where service provision times map perfectly and uniquely to payment amounts without gaps or repeat results for different inputs.
• Inverse Implication: If a function is bijective, it is guaranteed to have an inverse. This implies the function can "reverse," allowing us to go back from any earning to discover a distinct work period that corresponds to it.

Through confirming injectivity and surjectivity, we concluded the function is bijective. Consequently, it possesses an inverse, meaning from any provided output (like earning), we could trace it back to a specific input (time worked), reflecting a precise one-to-one mapping.