Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. y=-6x+4

Verified step by step guidance

1. Identify the given relation: \(y = -6x + 4\). This is an equation expressing \(y\) explicitly in terms of \(x\).
2. Determine if \(y\) is a function of \(x\): Since for each value of \(x\) there is exactly one corresponding value of \(y\) (because the equation is linear and passes the vertical line test), \(y\) is indeed a function of \(x\).
3. Find the domain: Since there are no restrictions on \(x\) in the equation \(y = -6x + 4\), the domain is all real numbers, which can be written as \((-\infty, \infty)\).
4. Find the range: Because the function is linear with a nonzero slope, \(y\) can take any real value as \(x\) varies over all real numbers. Therefore, the range is also \((-\infty, \infty)\).
5. Summarize: The relation defines \(y\) as a function of \(x\) with domain \((-\infty, \infty)\) and range \((-\infty, \infty)\).

Key Concepts

Definition of a Function

A function is a relation where each input (x-value) corresponds to exactly one output (y-value). To determine if y is a function of x, check that no x-value is paired with more than one y-value. For example, y = -6x + 4 is a function because for every x, there is a unique y.

Domain of a Function

The domain is the set of all possible input values (x-values) for which the function is defined. For linear functions like y = -6x + 4, the domain is typically all real numbers since any real x can be substituted without restriction.

Range of a Function

The range is the set of all possible output values (y-values) that the function can produce. For linear functions with nonzero slope, like y = -6x + 4, the range is all real numbers because as x varies over all real numbers, y takes on all real values.