Domanda del libro di testo
In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x2+x−6)/(x−3)
629
views
Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 4, Problema 83
In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x2+1)/x
Guida verificata passo dopo passo1
Identify the given rational function: \(f(x) = \frac{x^{2} + 1}{x}\).
To find the slant (oblique) asymptote, perform polynomial long division of the numerator by the denominator: divide \(x^{2} + 1\) by \(x\).
Set up the division: \(x\) divides into \(x^{2}\) exactly \(x\) times. Multiply \(x\) by \(x\) to get \(x^{2}\), subtract this from \(x^{2} + 1\) to find the remainder.
The remainder after subtracting is \(1\). So, the division gives \(x\) with a remainder of \(1\), which can be written as \(f(x) = x + \frac{1}{x}\).
The slant asymptote is the quotient without the remainder term, so it is the line \(y = x\). This line describes the behavior of \(f(x)\) as \(x\) approaches infinity or negative infinity.
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
16mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Rational Functions
A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x), where Q(x) ≠ 0. Understanding the behavior of rational functions, including their domains and asymptotes, is essential for graphing and analyzing their properties.
Video consigliato:
6:04
Intro to Rational Functions
Slant (Oblique) Asymptotes
Slant asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. They represent the line that the graph approaches as x approaches infinity or negative infinity, found by performing polynomial division.
Video consigliato:
6:24Introduction to Asymptotes
Polynomial Division
Polynomial division is a method used to divide one polynomial by another, similar to long division with numbers. It helps find the quotient and remainder, which are used to determine slant asymptotes and simplify rational functions for graphing.
Video consigliato:
05:13Introduction to Polynomials
Pratica correlata
Domanda del libro di testo
Exercises 82–84 will help you prepare for the material covered in the next section. Let f(x)=an(x4−3x2−4). If f(3)=−150, determine the value of a_n.
370
views
Domanda del libro di testo
In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x2−1)/x
648
views
Domanda del libro di testo
Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x2+4x+6=0
359
views
Domanda del libro di testo
Solve each inequality in Exercises 86–91 using a graphing utility. x2 + 3x - 10 > 0
573
views
Domanda del libro di testo
Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x2+4x−1=0
474
views