Problem 6 - 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? $$ \\{(0,3),(-1,5),(-2,7),(-3,9),(-4,11)\\} $$

Step-by-Step Solution

Answer

Domain: \(\{0, -1, -2, -3, -4\}\); Range: \(\{3, 5, 7, 9, 11\}\); One-to-One: Yes.

Step 1: Identify Domain

The domain of a function is the set of all possible input values (usually \(x\)-coordinates) that the function can accept. Looking at the set, the input values are \(0, -1, -2, -3, -4\). Thus, the domain is \(\{0, -1, -2, -3, -4\}\).

Step 2: Identify Range

The range of a function is the set of all possible output values (usually \(y\)-coordinates) that the function can produce. From the set, the output values are \(3, 5, 7, 9, 11\). Hence, the range is \(\{3, 5, 7, 9, 11\}\).

Step 3: Determine if Function is One-to-One

A function is one-to-one if each input value pairs with a unique output value and each output value is paired with a unique input value (no repeated \(y\) values for different \(x\) values). In this set, each input \(x\) value pairs with a unique output \(y\) value, and there are no repeated \(y\) values for different \(x\) values. Therefore, this function is one-to-one.

Key Concepts

Domain

The domain of a function refers to the complete set of possible input values, often represented as the set of all possible values of the independent variable, usually denoted as \( x \). In a function, these input values correspond to the first elements of ordered pairs. For the given set of ordered pairs \( \{(0,3),(-1,5),(-2,7),(-3,9),(-4,11)\} \), the inputs are \( 0, -1, -2, -3, \) and \( -4 \). Thus, the domain is the set \( \{0, -1, -2, -3, -4\} \).

• It's important to identify the domain when working with functions, as it defines the scope of inputs the function can accept.
• The domain determines the limitations or breadth of the function's applicability.

Range

The range of a function consists of all possible outputs or results provided by the function, often represented as the potential values of the dependent variable, usually denoted as \( y \). In a function defined by a set of ordered pairs, the range corresponds to the second elements of those pairs. Referring back to our set \( \{(0,3),(-1,5),(-2,7),(-3,9),(-4,11)\} \), the outputs are \( 3, 5, 7, 9, \) and \( 11 \). Therefore, the range is \( \{3, 5, 7, 9, 11\} \).

• Understanding the range helps to know what outputs we can expect from the function for a given domain.
• The range allows us to explore the function's behavior in response to its domain.

One-to-One Function

A one-to-one function is a special type of function where each input value is associated with a unique output value, and each output value corresponds to a unique input value. This ensures that there are no repeated output values (no two distinct input values have the same output).
Consider the function represented by the set \( \{(0,3),(-1,5),(-2,7),(-3,9),(-4,11)\} \). In this set, every input value maps to a distinct output value, and there are no duplicate output values for different input values. Therefore, this function is classed as one-to-one.

• A one-to-one function ensures that the function is invertible, meaning it has an inverse function that can reverse the mapping.
• This concept is crucial in various fields of mathematics, including algebra and calculus, for solving equations and analyzing functions.