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

Verified step by step guidance

1. Rewrite the inequality to express y in terms of x. Starting with \(x - y < 4\), isolate \(y\) by subtracting \(x\) from both sides: \(-y < 4 - x\).
2. Multiply both sides of the inequality by \(-1\) to solve for \(y\). Remember to reverse the inequality sign when multiplying by a negative number: \(y > x - 4\).
3. Analyze whether \(y\) is a function of \(x\). Since for each \(x\) there are infinitely many \(y\) values satisfying \(y > x - 4\), this relation does not define \(y\) as a function of \(x\) (a function must assign exactly one \(y\) to each \(x\)).
4. Determine the domain of the relation. Since there are no restrictions on \(x\) in the inequality, the domain is all real numbers, expressed as \((-\infty, \infty)\).
5. Determine the range of the relation. For each \(x\), \(y\) can be any value greater than \(x - 4\), so the range is also all real numbers, \((-\infty, \infty)\).

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 if for every x-value there is only one y-value. If any x maps to multiple y-values, the relation is not a function.

Solving Inequalities for y

To analyze the relation x - y < 4, solve the inequality for y to express it explicitly in terms of x. This helps identify the possible y-values for each x and determine if y is uniquely defined. Rearranging gives y > x - 4.

Domain and Range of Relations

The domain is the set of all possible x-values, and the range is the set of all possible y-values that satisfy the relation. For inequalities, the domain and range often include intervals or all real numbers, depending on the constraints.