In Exercises 11–26, determine whether each equation defines y as a function of x. |x|- y = 5

Verified step by step guidance

1. Step 1: Start with the given equation: |x| - y = 5.
2. Step 2: Solve for y in terms of x by isolating y on one side of the equation. Add y to both sides to get |x| = y + 5.
3. Step 3: Subtract 5 from both sides to solve for y: y = |x| - 5.
4. Step 4: Recall the definition of a function: A relation is a function if each input (x-value) corresponds to exactly one output (y-value).
5. Step 5: Analyze the equation y = |x| - 5. For each x-value, there is exactly one y-value because the absolute value function |x| is defined for all real numbers and outputs a single non-negative value. Therefore, y is a function of x.

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 creates a V-shaped graph, which can lead to multiple outputs for a single input when solving equations. Understanding how absolute value affects the relationship between x and y is essential for determining if the given equation defines y as a function of x.