Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. y=2/(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 expression is undefined), there is exactly one value of \(y\). Since the expression gives one output \(y\) for each input \(x\) (except where the denominator is zero), it defines \(y\) as a function of \(x\).
3. Find the domain: The domain consists of all real numbers \(x\) except where the denominator is zero. Set the denominator equal to zero and solve: \(x - 3 = 0 \implies x = 3\). So, the domain is all real numbers except \(x = 3\).
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 cannot reach. To find values \(y\) cannot take, set \(y = 0\) and solve for \(x\): \$0 = \frac{2}{x - 3}\(, which has no solution. So, \)y\( can be any real number except \)0$.
5. Summarize: The relation defines \(y\) as a function of \(x\) with domain \(\{x \in \mathbb{R} \mid x \neq 3\}\) and range \(\{y \in \mathbb{R} \mid y \neq 0\}\).

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 no x-value maps to multiple y-values. This ensures the relation passes the vertical line test.

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, since division by zero is undefined.

Range of a Function

The range is the set of all possible output values (y-values) the function can produce. For y = 2/(x-3), the range includes all real numbers except the value y cannot take due to the function's behavior, often found by analyzing limits or solving for x.