Problem 5 - Algebra 2 and Trigonometry

Question

In \(3-6,\) each set represents a function. a. What is the domain of each function? b. What is the range of each function? c.Is the function one-to-one? $$ \\{(2,7),(3,7),(4,7),(5,7),(6,7)\\} $$

Step-by-Step Solution

Answer

Domain: \{2, 3, 4, 5, 6\}; Range: \{7\}; Not one-to-one.

Step 1: Identify the domain

The domain of a function consists of all the first elements from each ordered pair. From the set \(\{(2,7),(3,7),(4,7),(5,7),(6,7)\}\), the domain is \(\{2, 3, 4, 5, 6\}\).

Step 2: Identify the range

The range of a function consists of all the second elements from each ordered pair. In this set, all second elements are \(7\), so the range is \(\{7\}\).

Step 3: Determine if the function is one-to-one

A function is one-to-one if each element of the domain maps to a unique element in the range. Here, all elements of the domain map to the same element \(7\) in the range. Therefore, the function is not one-to-one.

Key Concepts

Domain

In mathematics, the domain of a function is simply the set of all possible inputs that the function can accept. For functions depicted as sets of ordered pairs, the domain is made up of the first elements of these pairs. For instance, consider the set \(\{(2,7),(3,7),(4,7),(5,7),(6,7)\}\). Here, the domain is \(\{2, 3, 4, 5, 6\}\).
This means the function can accept and work with these specific numbers as inputs.
Understanding the domain is crucial, as it tells us the scope of numbers we can plug into the function without causing errors.

Range

While the domain focuses on the inputs, the range of a function is all about the possible outputs. To identify the range in a set of ordered pairs, look at all the second elements across the pairs.
In the example set \(\{(2,7),(3,7),(4,7),(5,7),(6,7)\}\), the range is simply \(\{7\}\) since every pair ends in 7.
This tells us that no matter which number from the domain you input into the function, the output will always be 7. Knowing the range is important for understanding what kind of results you can expect from the function.

One-to-One Functions

A one-to-one function is a special type of function where each element of the domain maps to a unique element in the range. In other words, no two different inputs should map to the same output.
To decide if a function is one-to-one, examine whether each input (from the domain) has a distinct output (in the range).
In the provided set \(\{(2,7),(3,7),(4,7),(5,7),(6,7)\}\), all the input values (2, 3, 4, 5, 6) produce the same output (7).
This means the function is not one-to-one because multiple inputs are linked to the same result. Identifying whether a function is one-to-one is essential, especially when considering inverse functions and ensuring that each element truly corresponds to a single, unique output.