In Exercises 11–26, determine whether each equation defines y as a function of x. |x| − y = 2

Verified step by step guidance

1. Rewrite the given equation |x| − y = 2 to isolate y. Subtract |x| from both sides: -y = 2 - |x|.
2. Multiply through by -1 to solve for y: y = |x| - 2.
3. Recall the definition of a function: A relation is a function if every input (x) corresponds to exactly one output (y).
4. Analyze the equation y = |x| - 2. The absolute value function |x| is defined for all real numbers x and always produces a single value. Subtracting 2 does not change this property.
5. Conclude that the equation y = |x| - 2 defines y as a function of x because for every x, there is exactly one corresponding value of y.

Key Concepts

Function Definition

A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. In mathematical terms, for a relation to be a function, no two ordered pairs can have the same first element with different second elements. This concept is crucial for determining if an equation defines y as a function of x.

Vertical Line Test

The vertical line test is a visual way to determine if a curve is a graph of a function. If any vertical line intersects the graph at more than one point, the relation is not a function. This test helps in analyzing equations graphically to see if they meet the criteria of a function.

Absolute Value Function

The absolute value function, denoted as |x|, outputs the non-negative value of x regardless of its sign. This function can produce two outputs for a single input when set equal to another variable, which can complicate the determination of whether y is a function of x. Understanding how absolute values behave is essential for analyzing the given equation.