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. Step 1: Understand the definition of a function. A relation defines y as a function of x if for every x-value, there is exactly one corresponding y-value.
2. Step 2: Analyze the given equation y = -\frac{7}{x-5}. Notice that the equation is in the form of a rational function.
3. Step 3: Identify any restrictions on the domain. The denominator x-5 cannot be zero, as division by zero is undefined. Therefore, x cannot be 5.
4. Step 4: Determine the domain of the function. The domain is all real numbers except x = 5, which can be expressed as (-\infty, 5) \cup (5, \infty).
5. Step 5: Determine the range of the function. Since the function is a rational function and the numerator is a constant, the range is all real numbers except y = 0, which can be expressed as (-\infty, 0) \cup (0, \infty).

Key Concepts

Function Definition

A function is a relation where each input (x-value) corresponds to exactly one output (y-value). To determine if a relation defines y as a function of x, we check if any x-value is paired with more than one y-value. For example, the relation y = -7/(x-5) is a function because for every x (except x=5), there is a unique y.

Domain

The domain of a function is the set of all possible input values (x-values) that can be used without causing any mathematical issues, such as division by zero. In the case of y = -7/(x-5), the domain excludes x = 5, where the function is undefined, leading to the domain being all real numbers except 5.

Range

The range of a function is the set of all possible output values (y-values) that the function can produce. For the function y = -7/(x-5), as x approaches 5, y approaches negative or positive infinity, but it never actually reaches zero. Thus, the range is all real numbers except for y = 0.