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

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Understand the problem: We need to determine if the relation defined by the inequality \(x - y < 4\) defines \(y\) as a function of \(x\). Then, we will find the domain and range of this relation.

Rewrite the inequality to express \(y\) in terms of \(x\): Starting from \(x - y < 4\), subtract \(x\) from both sides to get \(-y < 4 - x\). Then multiply both sides by \(-1\) (remember to reverse the inequality sign) to get \(y > x - 4\).

Analyze if \(y\) is a function of \(x\): For each fixed value of \(x\), check if there is exactly one corresponding value of \(y\). Since \(y > x - 4\) represents all \(y\) values greater than \(x - 4\), there are infinitely many \(y\) values for each \(x\). Therefore, \(y\) is not a function of \(x\) because a function must assign exactly one output for each input.

Determine the domain: Since there is no restriction on \(x\) in the inequality, the domain is all real numbers, which can be written as \((-\infty, \infty)\).

Determine the range: For each \(x\), \(y\) can be any value greater than \(x - 4\). Since \(x\) can be any real number, the smallest possible lower bound for \(y\) can be arbitrarily large negative or positive. Hence, the range is also all real numbers, \((-\infty, \infty)\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

Inequalities and Graphical Representation

The inequality x - y < 4 describes a region in the coordinate plane. Understanding how to interpret and graph inequalities helps visualize the relation and analyze whether it defines y as a function of x by checking vertical line intersections.

Domain and Range of a Relation

The domain is the set of all possible x-values, and the range is the set of all possible y-values in the relation. Identifying these sets involves analyzing the inequality and determining which x and y satisfy it.