Skip to main content
Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblema 35
Capitolo 4, Problema 35

Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. r(x)=(x2+4x−21)/(x+7)

Guida verificata passo dopo passo
1
Start by identifying the rational function given: \(r(x) = \frac{x^{2} + 4x - 21}{x + 7}\).
Factor the numerator \(x^{2} + 4x - 21\) to see if any factors cancel with the denominator. To factor, find two numbers that multiply to \(-21\) and add to \(4\).
Write the factored form of the numerator and check if the denominator \(x + 7\) is a factor of the numerator. If it is, this indicates a hole at the value of \(x\) that makes \(x + 7 = 0\).
If the factor cancels, the hole occurs at \(x = -7\). If it does not cancel, then \(x = -7\) is a vertical asymptote because the denominator is zero there and the function is undefined.
Summarize: vertical asymptotes occur where the denominator is zero and not canceled by the numerator, and holes occur where a factor cancels between numerator and denominator.

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:
2m

Concetti 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). Understanding the behavior of rational functions involves analyzing their numerators and denominators, especially where the denominator equals zero, which affects the domain and graph.
Video consigliato:
6:04
Intro to Rational Functions

Vertical Asymptotes

Vertical asymptotes occur at values of x where the denominator of a rational function is zero and the numerator is nonzero, causing the function to approach infinity or negative infinity. Identifying these points helps describe the function's behavior near undefined values.
Video consigliato:
3:12
Determining Vertical Asymptotes

Holes in the Graph

Holes occur when a factor cancels out from both numerator and denominator, indicating a removable discontinuity. At these x-values, the function is undefined, but the limit exists, resulting in a 'hole' rather than an asymptote on the graph.
Video consigliato:
3:34
Determining Removable Discontinuities (Holes)
Pratica correlata
Domanda del libro di testo

In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x4+6x3−18x2; between 2 and 3

1187
views
Domanda del libro di testo

Use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=x4−6x3+14x2−14x+5f(x) = x^4 - 6x^3 + 14x^2 -14x + 5

1049
views
Domanda del libro di testo

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x3+2x2−4x−8≥0x^3+2x^2−4x−8≥0

512
views
Domanda del libro di testo

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x3+x2−2x+1; between -3 and -2

863
views
Domanda del libro di testo

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=5x3−3x2+3x−1

464
views
Domanda del libro di testo

Use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=3x3−7x2−2x+5;f(−3)

451
views