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

Verified step by step guidance

1. Identify the given relation: \(y = \frac{2}{x - 3}\). This is a rational function where \(y\) depends on \(x\).
2. Determine if \(y\) is a function of \(x\): For each value of \(x\) (except where the denominator is zero), there is exactly one value of \(y\). Since the denominator \(x - 3\) cannot be zero, \(x \neq 3\). Therefore, \(y\) is a function of \(x\).
3. Find the domain: The domain consists of all real numbers except where the denominator is zero. Set the denominator equal to zero and solve for \(x\): \(x - 3 = 0 \Rightarrow x = 3\). So, the domain is all real numbers \(x\) such that \(x \neq 3\).
4. Find the range: Consider the values that \(y\) can take. Since \(y = \frac{2}{x - 3}\), \(y\) can be any real number except where the function is undefined or does not reach. To find any restrictions, set \(y = 0\) and solve for \(x\): \(0 = \frac{2}{x - 3}\), which has no solution. So, \(y\) can never be zero. Hence, the range is all real numbers \(y\) such that \(y \neq 0\).
5. Summarize: The relation defines \(y\) as a function of \(x\) with domain \(\{x \in \mathbb{R} : x \neq 3\}\) and range \(\{y \in \mathbb{R} : y \neq 0\}\).

Key Concepts

Definition of a Function

A function is a relation where each input x corresponds to exactly one output y. To determine if y is a function of x, check that for every x-value, there is only one y-value. If any x maps to multiple y-values, the relation is not a function.

Domain of a Function

The domain is the set of all possible input values (x-values) for which the function is defined. For rational functions like y = 2 / (x - 3), the domain excludes values that make the denominator zero, as division by zero is undefined.

Range of a Function

The range is the set of all possible output values (y-values) the function can take. To find the range, analyze the behavior of the function and identify any values y cannot take, often by solving for x in terms of y and considering domain restrictions.