#12 can you show me how to do this right? I’ve attached my work. **Problem Statement:**Find all horizontal and vertical asymptotes (if any). (If an answer does not exist, enter DNE.)\[s(x) = \frac{16x^2 + 1}{4x^2 + 2x - 6}\]**Answers Provided:**- Vertical asymptote: \( x = 1 \) (smaller value) ❌- Vertical asymptote: \( x = -\frac{6}{4} \) (larger value) ❌- Horizontal asymptote: \( y = 16 \) ❌**Note:**The problem requires identifying the correct horizontal and vertical asymptotes of the given rational function. The incorrect answers are marked with red crosses (❌). The problem presented is:\[ s(x) = \frac{16x^2 + 1}{4x^2 + 2x - 6} \]**Solution Steps:**1. **Identify and Factor the Denominator:** - The task is to simplify the expression, starting by factoring the denominator: \( 4x^2 + 2x - 6 \).2. **Factorization Steps:** - Decompose into simpler expressions using cross multiplication indications: - Write down the product \( ac = 24 \) (from \( a = 4, c = -6 \)). - Find the factors of 24 which sum up to \( b = 2 \). These are \(-4\) and \(6\).3. **Intermediate Steps:** - Break the middle term using these factors: \[ 4x^2 - 4x + 6x - 6 \] - Factor by grouping: \[ 4x(x-1) + 6(x-1) \] - Combine common factors: \[ (x-1)(4x+6) \]4. **Vertical Asymptote (VA) Identification:** - Set the denominator equal to zero to find the vertical asymptotes. - \( x - 1 = 0 \) results in a vertical asymptote at \( x = 1 \).5. **Simplifying by Solving Remainder Equation:** - Solve the equation \( 4x + 6 = 0 \): \[ 4x = -6 \] - Simplify to find: \[ x = -\frac{6}{4} = -\frac{3}{2} \]**Conclusion:**The function has a vertical asymptote at \( x = 1 \). Further simplification might include reducing or canceling terms, if possible, depending on the form of \( s(x) \).

Answered: Find all horizontal and vertical…

Find all horizontal and vertical asymptotes (if any). (If an answer does not exist, enter DNE.) 16x2 + 1 s(x) = 4x2 %3D + 2x – 6

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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