Problem 129 - Algebra and Trigonometry

Question

Determine whether each statement is true or false If the statement is false, make the necessary change(s) to produce a true statement. The graph of a rational function can never cross a vertical asymptote.

Step-by-Step Solution

Answer

The statement 'The graph of a rational function can never cross a vertical asymptote' is true.

Step 1: Understanding Vertical asymptotes

Analyze and understand the concept of a vertical asymptote. Vertical asymptotes in a rational function occur at the x-values where the denominator of the function equals zero. A vertical asymptote is a vertical line where the function value approaches infinite as it gets closer to the x-coordinate of the line. Therefore, the function can never cross a vertical asymptote.

Step 2: Evaluate the Statement

Given the statement: 'The graph of a rational function can never cross a vertical asymptote.' If you apply the knowledge about vertical asymptotes of rational functions, then it becomes clear that this statement is indeed true. By definition, a rational function cannot intersect with its vertical asymptote as the function value approaches infinity or negative infinity as it gets closer to the vertical asymptote.

Key Concepts

Vertical Asymptotes

A vertical asymptote is a line that the graph of a function approaches but never actually touches or crosses as it heads towards infinity. In the case of a rational function, vertical asymptotes are present where the function's denominator equals zero. For instance, given a rational function in the form \( f(x) = \frac{p(x)}{q(x)} \), a vertical asymptote occurs wherever \( q(x) = 0 \) and \( p(x) \) is non-zero.

• This creates a vertical line at that x-value, known as the asymptote.
• Since the function values can become very large (positive or negative) without actually reaching the line, it forms a boundary that cannot be crossed.

Understanding this behavior is key in graphing rational functions as it shows points of undefined behavior for the function.

Rational Function Graph

Graphing a rational function involves looking for points where the function is undefined, which often leads to vertical asymptotes, and other features like horizontal asymptotes, intercepts, and behaviors at infinity. When mapping out these functions, you often see them approaching these boundaries but never crossing them.

• Vertical Asymptotes: Indicate undefined points, where the function spikes up or down.
• Horizontal Asymptotes: Dictate the function's behavior as x approaches infinity.
• Intercepts: Where the function crosses the x or y-axis.

By plotting these elements, you gain a full picture of the function’s behavior, especially highlighting where it increases without bounds or approaches a steady state value.

Asymptote Crossing

Asymptotes, particularly vertical ones, are lines that graphs of rational functions approach but do not cross. When a rational function has a vertical asymptote, this signifies that the function's value becomes infinitely large (either positively or negatively) as the graph approaches this line.

• Unlike vertical asymptotes, horizontal asymptotes may be intersected by the function's graph at finite points, although the function will align with them as \( x \) approaches infinity.
• An example is a rational function possibly crossing its horizontal asymptote before stabilizing.

Therefore, for vertical asymptotes, the statement holds true: the graph never crosses, adhering to its definition based on division by zero avoidance.