Question

What are the domain and range of the addition function on the real numbers? On Multiplication? Subtraction? Division?

Short answer

The domain and range for addition, multiplication, and subtraction are all real numbers; for division, the domain excludes dividing by zero, and the range is all real numbers except zero.

Step-by-step solution

Understanding the Problem

We are asked to find the domain and range of four different arithmetic operations when they are applied to real numbers: addition, multiplication, subtraction, and division.

Addition Function

The addition function takes two real numbers and returns another real number. Since there are no restrictions on what real numbers can be added, the domain of the addition function is all real numbers, and the range is also all real numbers.

Multiplication Function

Similarly, the multiplication function multiplies two real numbers and returns a real number. There are no restrictions on what real numbers can be multiplied, so the domain of the multiplication function is all real numbers, and the range is also all real numbers.

Subtraction Function

For the subtraction function, any real number can be subtracted from another real number. Therefore, the domain is all real numbers. Every possible real number can be the result of a subtraction, so the range is also all real numbers.

Division Function

The division function is defined for all real numbers except when dividing by zero. This means the domain includes all pairs of real numbers where the divisor is not zero. The range of the division function is all real numbers except zero, as there is no real number that results from dividing by zero.

Key concepts

Addition Function

The addition function is one of the most fundamental operations in mathematics. It involves combining two numbers to make a new, larger number. For example, if you add 3 and 5, you get 8. When considering the domain and range of the addition function on the real numbers, it's quite straightforward.

• Domain: The domain of the addition function includes all real numbers. You can add any real number to another, meaning there are no restrictions. Whether it's adding positives, negatives, fractions, or decimals, every real number works.
• Range: Likewise, the range is all real numbers. Whatever numbers you add together, the result will always be another real number.

In essence, the addition function can be expressed as \( f(x, y) = x + y \), where both \( x \) and \( y \) are any real numbers.

Multiplication Function

Much like addition, multiplication is an essential arithmetic operation where two numbers are combined to give a product. For example, multiplying 2 with 4 will result in 8. In terms of its domain and range, the multiplication function is also very inclusive.

• Domain: The domain encompasses all real numbers, meaning you can multiply any real number with another. Whether factors are positive, negative, or zero, every real number is applicable.
• Range: The range of multiplication covers all real numbers as well. The product of any two real numbers will always be another real number.

Thus, the multiplication function is defined as \( g(x, y) = x \, \cdot \, y \), with \( x \) and \( y \) being any real numbers.

Subtraction Function

Subtraction, an operation that finds the difference between two numbers, is just as fundamental. For example, subtracting 4 from 9 gives a result of 5. Subtraction as a function is similar in scope to addition and multiplication.

• Domain: Similar to the addition function, the domain of subtraction includes all real numbers. You can subtract any real number from another without restriction.
• Range: The range is also all real numbers, as the outcome of subtracting one real number from another is always a real number.

Therefore, the subtraction function can be expressed as \( h(x, y) = x - y \), where both \( x \) and \( y \) represent any real number.

Division Function

Division indicates splitting one number by another, and it is slightly more complex than addition, multiplication, or subtraction. For instance, dividing 10 by 2 results in 5. The domain and range of the division function require special consideration due to the division by zero problem.

• Domain: The domain of the division function is all real numbers except situations where the divisor is zero. You cannot divide by zero, so if the second number in the division, \( y \), is zero, the operation is undefined.
• Range: Similarly, the range excludes zero because no real number multiplied by zero can yield a nonzero number. This means any other real number can be a division result.

Therefore, the division function can be articulated as \( q(x, y) = \frac{x}{y} \), with the condition that \( y eq 0 \).