Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. y=-7/(x-5)

Verified step by step guidance

1. Identify the given relation: \(y = \frac{-7}{x - 5}\). This is a rational function where \(y\) is expressed in terms of \(x\).
2. Determine if \(y\) is a function of \(x\): For each value of \(x\) (except where the expression is undefined), there is exactly one corresponding value of \(y\). Since the expression gives a unique \(y\) for each \(x\) 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: \(x - 5 = 0 \implies x = 5\). So, the domain is all real numbers except \(x = 5\).
4. Find the range: Consider the values that \(y\) can take. Since \(y = \frac{-7}{x - 5}\), \(y\) can be any real number except where the function is undefined or approaches a horizontal asymptote. Analyze the behavior as \(x\) approaches 5 and as \(x\) approaches infinity to understand the range.
5. Summarize: Conclude that \(y\) is a function of \(x\) with domain \(\{x \in \mathbb{R} \mid x \neq 5\}\) and describe the range based on the behavior of the function.

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 when graphed.

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 = -7/(x-5), 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 produce. Finding the range often involves analyzing the behavior of the function and identifying any values y cannot take, such as horizontal asymptotes or restrictions from the function's formula.