Problem 95 - Algebra A Combined Function

Question

Think about the appearance of each graph. Without graphing, determine which equations represent functions. Explain each answer. \(y=2 x\)

Step-by-Step Solution

Answer

The equation y = 2x represents a function because for every x-value, there is a unique y-value.

Step 1: Understanding the Concept of a Function

A function is a relation where each input has a single, unique output. If an equation represents a function, any given x-value in its domain corresponds to exactly one y-value.

Step 2: Analyzing y = 2x

In this equation, y is explicitly defined in terms of x such that for every x-value, there is precisely one y-value determined by the expression y = 2x. This satisfies the criteria for a function.

Step 3: Graphical Interpretation

Thinking about the graph of y = 2x, it is a straight line with a slope of 2. A straight line can be graphed using the equation y = mx + c (where m = slope, c = y-intercept), and such lines are always functions unless they are vertical lines.

Step 4: Vertical Line Test

The vertical line test is an intuitive way to determine if a graph represents a function. A graph represents a function if and only if no vertical line can intersect the graph more than once at any given x-coordinate. Since y = 2x is a non-vertical line, it passes the vertical line test.

Key Concepts

Graph of a Linear Equation

A linear equation such as \( y = 2x \) describes a straight line on the coordinate plane. In this equation, each value of \( x \) is paired with exactly one value of \( y \), which can be expressed in the "slope-intercept form": \( y = mx + c \). Here, \( m \) represents the slope, which indicates how steep the line is, and \( c \) represents the y-intercept, where the line crosses the y-axis.

• In the equation \( y = 2x \), the slope \( m \) is 2 and the y-intercept \( c \) is 0.
• This means the line rises two units for every one unit it moves to the right.

Visualizing a graph can reveal that the line is straight, consistent with its linear nature. This consistency ensures that each x-value corresponds with one and only one y-value.

Vertical Line Test

The vertical line test is a simple way to determine whether a graph represents a function. This test consists of visualizing or drawing imaginary vertical lines across the graph. If any vertical line intersects the graph at more than one point, then the graph does not represent a function.

• For the linear equation \( y = 2x \), no vertical line can intersect the line more than once.
• Thus, it passes the vertical line test, satisfying the condition for being a function.

By passing the vertical line test, the line associated with \( y = 2x \) confirms that every input \( x \) has one unique output \( y \). This aligns with the definition of a function.

Equation Representation as a Function

An equation represents a function if for every input value, there is exactly one output value. For linear equations, as long as the graph of the line is not vertical, the equation acts as a function.

• In the equation \( y = 2x \), each \( x \) directly determines a single \( y \) value.
• This direct relationship ensures that \( y = 2x \) is a function.

The characteristic that distinguishes a function is this unique correspondence between inputs and outputs. Lines not vertical in nature inherently provide this singular connection between \( x \) and \( y \), ensuring a functional representation.