Problem 13 - Algebra 2 and Trigonometry

Question

In \(12-23,\) each set is a function from set \(A\) to set \(B .\) a. What is the largest subset of the real numbers that can be set \(A\) , the domain of the given function? b. If set \(A=\operatorname{set} B,\) is the function onto? Justify your answer. $$ \\{(x, y) : y=5-x\\} $$

Step-by-Step Solution

Answer

a. Set \( A \) is \( \mathbb{R} \). b. Yes, the function is onto if \( A = B = \mathbb{R} \).

Step 1: Understand the Function Representation

We have the function defined as \( \{(x, y) : y = 5 - x \} \). This means for each input value \( x \), the output \( y \) is calculated by the expression \( 5 - x \). This describes a linear relationship between \( x \) and \( y \).

Step 2: Determine the Domain of the Function

To identify the largest possible set for the domain set \( A \), we consider the context of real numbers. Here, \( x \) can be any real number because the expression \( 5 - x \) is defined for all real numbers. Thus, the largest subset of real numbers that can be set \( A \) is the set of all real numbers, \( \mathbb{R} \).

Step 3: Define Onto Function

A function is onto (surjective) if every element in set \( B \) has a corresponding element in set \( A \). For this function, since \( y = 5 - x \), we need to determine if for every real number \( y \) there exists a real number \( x \) such that \( y = 5 - x \).

Step 4: Check if Function is Onto

For any real number \( y \), you can find an \( x \) by rearranging the equation to \( x = 5 - y \). Since both \( x \) and \( y \) can take any real value, this function is onto when \( A = B = \mathbb{R} \). Each \( y \) value in set \( B \) is accounted for by some \( x \) value in set \( A \).

Key Concepts

Domain and Range

In the context of functions, the domain refers to the set of all possible input values that the function can accept. For the function given as \( \{(x, y) : y = 5 - x \} \), the domain is determined by considering all possible real numbers for \( x \). This is because the expression \( 5 - x \) can handle any real number input. Therefore, the domain of this function is the set of all real numbers, denoted as \( \mathbb{R} \).
To understand the range, we need to look at all possible output values of \( y \). Since \( y = 5 - x \), and \( x \) is any real number, \( y \) will also span all real numbers. As a result, the range of this function is also \( \mathbb{R} \).
In summary:

• Domain: all real numbers (\( \mathbb{R} \)).
• Range: all real numbers (\( \mathbb{R} \)).

Onto Function

An onto function, also known as a surjective function, ensures that every element in the codomain (set \( B \)) is mapped by at least one element from the domain (set \( A \)). For the given function \( y = 5 - x \), we need to see if every possible \( y \) value in set \( B \) can be produced by some \( x \) value from set \( A \).
To check this, consider: for any given real number \( y \), does there exist a real number \( x \) such that \( y = 5 - x \)? We can rearrange this as \( x = 5 - y \).
Since both \( x \) and \( y \) can potentially be any real number, for any value of \( y \) you select, you can find a corresponding \( x \) value, confirming that every output in the codomain is hit by the function. Thus, the function is onto when set \( A = B = \mathbb{R} \).
Key characteristics of an onto function include:

• Every element of the codomain is mapped by some element of the domain.
• It covers the entire range of possible output values.

Linear Equations

Linear equations are equations of the first degree, meaning they involve no powers higher than one. Such equations form straight lines when graphed. The standard form of a linear equation in two variables is \( ax + by = c \).
In our given function \( y = 5 - x \), the equation is already in the form of \( y = mx + b \), where:

• \( m \) is the slope (-1 in this case).
• \( b \) is the y-intercept (5 here).

Linear equations like this one describe a straight line where every point on the line represents a solution to the equation. The slope \( m \) describes the steepness and direction of the line. A positive slope means the line rises, while a negative slope means it falls as it moves from left to right.
These functions are fundamental in mathematics because they model relations where changes are consistent. They're used extensively in applications that involve constant rates, such as velocity in physics or cost calculations in economics.