Determine whether each relation defines y as a function of x. Give the domain and range. 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 is a fraction with \(x\) in the denominator, check where the denominator is zero.
3. Find the domain by setting the denominator not equal to zero: Solve \(x - 5 \neq 0\), which gives \(x \neq 5\). So, the domain is all real numbers except \(x = 5\).
4. Since for each \(x\) in the domain there is exactly one \(y\), the relation defines \(y\) as a function of \(x\).
5. Determine the range: Consider the values \(y\) can take. Since \(y = \frac{-7}{x - 5}\), \(y\) can be any real number except possibly some value that makes the expression undefined or impossible. Analyze the behavior of the function to describe the range.

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 = -7/(x-5), 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) that the function can produce. To find the range, analyze the behavior of the function and determine which y-values are attainable given the domain restrictions.