Determine whether each relation defines y as a function of x. Give the domain and range. y=2/(x-3)

Verified step by step guidance

1. Identify the given relation: \(y = \frac{2}{x - 3}\).
2. Determine if \(y\) is a function of \(x\): For each value of \(x\) (except where the expression is undefined), there should be exactly one corresponding value of \(y\). Since the expression is a fraction with \(x\) in the denominator, check where the denominator equals zero.
3. Find the values of \(x\) that make the denominator zero by solving \(x - 3 = 0\), which gives \(x = 3\). This value is excluded from the domain because division by zero is undefined.
4. State the domain: All real numbers except \(x = 3\), which can be written as \(\{x \in \mathbb{R} \mid x \neq 3\}\).
5. Determine the range: Since \(y = \frac{2}{x - 3}\) can take any real value except possibly some value that \(y\) cannot equal, analyze if there are any restrictions on \(y\). Because the function is a rational function with a vertical asymptote at \(x=3\) and no horizontal asymptote restricting \(y\), the range is all real numbers except \(y = 0\) (since \(\frac{2}{x-3} = 0\) has no solution).

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), analyze the behavior of the function and any horizontal asymptotes to determine which y-values are attainable.