Math Problem Statement

how do I solve this?

How do I solve this?

Solution

Let's analyze the problem step-by-step:

(i) Identify the horizontal and vertical asymptotes (if any). Explain how you would find them mathematically.

  1. Vertical Asymptotes:
    Vertical asymptotes occur when the denominator of the rational function equals zero (and the numerator does not cancel it out). From the graph:

    • The vertical asymptotes appear at x=0x = 0 and x=4x = 4, as the graph approaches infinity near these xx-values.
  2. Horizontal Asymptotes:
    Horizontal asymptotes depend on the degrees of the numerator (nn) and denominator (mm) of the rational function:

    • If n<mn < m, the horizontal asymptote is y=0y = 0.
    • If n=mn = m, the horizontal asymptote is y=leading coefficient of numeratorleading coefficient of denominatory = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}.
    • If n>mn > m, there is no horizontal asymptote.

    From the graph, as xx \to \infty, the curve approaches y=1y = 1, indicating a horizontal asymptote at y=1y = 1.


(ii) Identify the zeros of the rational function.

Zeros occur when the numerator equals zero (and the denominator is non-zero). From the graph:

  • The zeros of the function are x=2x = 2 and x=3x = 3, as these are the points where the graph crosses the xx-axis (y=0y = 0).

(iii) Identify the rational function.

To construct the rational function:

  1. Zeros: The numerator must have factors corresponding to the zeros at x=2x = 2 and x=3x = 3. Hence, the numerator is (x2)(x3)(x - 2)(x - 3).
  2. Vertical Asymptotes: The denominator must have factors corresponding to the vertical asymptotes at x=0x = 0 and x=4x = 4. Hence, the denominator is x(x4)x(x - 4).
  3. Horizontal Asymptote: The horizontal asymptote is y=1y = 1. This implies the degrees of the numerator and denominator are the same, and the leading coefficients of both are equal. Therefore, no additional constants are needed.

The rational function is: f(x)=(x2)(x3)x(x4).f(x) = \frac{(x - 2)(x - 3)}{x(x - 4)}.


Would you like me to expand on the derivations or clarify any part? Here are related questions to deepen understanding:

  1. How do you confirm that y=1y = 1 is the horizontal asymptote mathematically?
  2. Why do vertical asymptotes occur at x=0x = 0 and x=4x = 4?
  3. What would happen if a zero and an asymptote occur at the same xx-value?
  4. How can you verify the zeros x=2x = 2 and x=3x = 3 satisfy the equation f(x)=0f(x) = 0?
  5. How would the graph change if there were an additional vertical asymptote?

Tip: Always verify the symmetry and behavior near asymptotes when sketching or analyzing rational functions.

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Math Problem Analysis

Mathematical Concepts

Rational Functions
Asymptotes (Vertical and Horizontal)
Zeros of Functions

Formulas

Horizontal asymptote rules: y = 0 if n < m, y = leading coefficient ratio if n = m, no horizontal asymptote if n > m
Vertical asymptotes occur when the denominator equals zero

Theorems

Properties of Rational Functions

Suitable Grade Level

Grades 10-12