Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. 2y = ——— x - 3

Verified step by step guidance

1. Identify the given equation: \( y = \frac{2}{x - 3} \).
2. Determine if the equation defines \( y \) as a function of \( x \). A relation is a function if each input \( x \) has exactly one output \( y \).
3. Notice that the equation is a rational function, which is defined for all real numbers except where the denominator is zero.
4. Find the value of \( x \) that makes the denominator zero: set \( x - 3 = 0 \) and solve for \( x \).
5. The domain of the function is all real numbers except \( x = 3 \). The range is all real numbers except \( y = 0 \), as the function never equals zero.

Key Concepts

Function Definition

A function is a relation where each input (x-value) corresponds to exactly one output (y-value). To determine if a relation defines y as a function of x, we check if any x-value is paired with more than one y-value. In this case, the equation y = 2/(x - 3) must be analyzed to see if it meets this criterion.

Domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the given relation y = 2/(x - 3), the function is undefined when the denominator equals zero, which occurs at x = 3. Therefore, the domain excludes this value, resulting in all real numbers except x = 3.

Range

The range of a function is the set of all possible output values (y-values) that the function can produce. For the relation y = 2/(x - 3), as x approaches 3, y approaches infinity or negative infinity, but never actually reaches zero. Thus, the range includes all real numbers except y = 0, indicating that the function can take on any value except for zero.