Math Problem Statement

Determine the domain, range, and asymptotes for the functions: y = (2x - 5) / (2x + 5) and y = (x^2 - 1) / (2x^2 - 3x - 2).

Solution

The image contains two rational functions for which we need to determine the domain, range, and asymptotes.

Function 1: y=2x52x+5y = \frac{2x - 5}{2x + 5}

  1. Domain: The domain of a rational function is all real numbers except where the denominator is zero. 2x+5=0x=522x + 5 = 0 \quad \Rightarrow \quad x = -\frac{5}{2} So, the domain is xR{52}x \in \mathbb{R} \setminus \left\{ -\frac{5}{2} \right\}.

  2. Range: To find the range, we can use the fact that horizontal asymptotes can give insight into the behavior of the function as x±x \to \pm \infty. Since the degrees of the numerator and denominator are the same, the horizontal asymptote is determined by the ratio of the leading coefficients: Horizontal Asymptote: y=22=1\text{Horizontal Asymptote: } y = \frac{2}{2} = 1 However, since the function never actually reaches the horizontal asymptote but can approach any value above or below 1, the range is yR{1}y \in \mathbb{R} \setminus \{ 1 \}.

  3. Asymptotes:

    • Vertical Asymptote: Occurs where the denominator is zero, which is x=52x = -\frac{5}{2}.
    • Horizontal Asymptote: y=1y = 1.

Function 2: y=x212x23x2y = \frac{x^2 - 1}{2x^2 - 3x - 2}

  1. Domain: We need to find where the denominator equals zero to exclude these values from the domain. 2x23x2=02x^2 - 3x - 2 = 0 Solving this quadratic equation using the quadratic formula: x=(3)±(3)24(2)(2)2(2)=3±9+164=3±254x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(-2)}}{2(2)} = \frac{3 \pm \sqrt{9 + 16}}{4} = \frac{3 \pm \sqrt{25}}{4} x=3±54x=2orx=12x = \frac{3 \pm 5}{4} \quad \Rightarrow \quad x = 2 \quad \text{or} \quad x = -\frac{1}{2} So, the domain is xR{2,12}x \in \mathbb{R} \setminus \{ 2, -\frac{1}{2} \}.

  2. Range: Finding the range analytically for more complex rational functions is challenging, but we can use a numerical or graphical approach. Generally, it involves investigating the behavior of the function as x±x \to \pm \infty.

  3. Asymptotes:

    • Vertical Asymptotes: These occur where the denominator is zero, which gives vertical asymptotes at x=2x = 2 and x=12x = -\frac{1}{2}.
    • Horizontal Asymptote: Since the degrees of the numerator and denominator are equal, the horizontal asymptote is determined by the ratio of the leading coefficients: y=12y = \frac{1}{2}

Summary:

  • For y=2x52x+5y = \frac{2x - 5}{2x + 5}:

    • Domain: xR{52}x \in \mathbb{R} \setminus \left\{ -\frac{5}{2} \right\}
    • Range: yR{1}y \in \mathbb{R} \setminus \{ 1 \}
    • Vertical Asymptote: x=52x = -\frac{5}{2}
    • Horizontal Asymptote: y=1y = 1
  • For y=x212x23x2y = \frac{x^2 - 1}{2x^2 - 3x - 2}:

    • Domain: xR{2,12}x \in \mathbb{R} \setminus \{ 2, -\frac{1}{2} \}
    • Range: Needs further analysis or graphical investigation.
    • Vertical Asymptotes: x=2x = 2 and x=12x = -\frac{1}{2}
    • Horizontal Asymptote: y=12y = \frac{1}{2}

Let me know if you'd like further details or a graphical analysis!

Here are five related questions to expand your understanding:

  1. How do we determine slant (oblique) asymptotes for rational functions?
  2. What methods can be used to find the range of a rational function besides graphical approaches?
  3. How do horizontal asymptotes affect the long-term behavior of a rational function?
  4. What are removable discontinuities in rational functions, and how can they be identified?
  5. How does the degree of the numerator and denominator influence the behavior of a rational function?

Tip: Always check for common factors in the numerator and denominator before determining vertical asymptotes. Sometimes a factor can cancel, resulting in a removable discontinuity instead of an asymptote.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Rational Functions
Asymptotes
Domain and Range

Formulas

Domain: Set where the denominator is not equal to zero.
Range: Horizontal asymptotes and function behavior as x approaches infinity.
Vertical Asymptotes: Occur where the denominator is zero and the function is undefined.
Horizontal Asymptote: Ratio of leading coefficients for same-degree polynomials.

Theorems

Quadratic Formula
Asymptote Properties

Suitable Grade Level

Grade 11-12