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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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