In Exercises 55–64, use the vertical line test to identify graphs in which y is a function of x.

Verified step by step guidance

1. Step 1: Understand the vertical line test. The vertical line test states that a graph represents a function if and only if no vertical line intersects the graph at more than one point.
2. Step 2: Observe the given graph. Notice the shape and orientation of the curve.
3. Step 3: Imagine or draw vertical lines at various points along the x-axis. For example, draw vertical lines at x = 0, x = 5, x = 10, etc.
4. Step 4: Check the intersections of these vertical lines with the graph. If any vertical line intersects the graph at more than one point, then y is not a function of x.
5. Step 5: Conclude whether the graph passes the vertical line test. If all vertical lines intersect the graph at most once, then y is a function of x. Otherwise, it is not.

Key Concepts

Vertical Line Test

The vertical line test is a method used to determine if a curve on a graph represents a function. If any vertical line intersects the graph at more than one point, the graph does not represent a function of x. This test is essential for identifying whether y is a function of x, as it ensures that each input (x-value) corresponds to exactly one output (y-value).

Function Definition

A function is defined as 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. Understanding this definition is crucial for analyzing graphs and determining if they represent functions.

Graph Interpretation

Graph interpretation involves analyzing the visual representation of data or equations to extract meaningful information. In the context of functions, it requires understanding how the shape and behavior of a graph relate to the function's properties, such as continuity, limits, and the relationship between variables. This skill is vital for applying the vertical line test effectively.