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 61
Capitolo 4, Problema 61

Find the domain of each function. f(x)=2x2−5x+2f(x) = \(\sqrt{2x^2 - 5x + 2}\)

Guida verificata passo dopo passo
1
Identify the function given: \(f(x) = \sqrt{2x^2 - 5x + 2}\). Since this is a square root function, the expression inside the square root must be greater than or equal to zero for the function to be defined.
Set up the inequality for the radicand (the expression inside the square root): \(2x^2 - 5x + 2 \geq 0\).
Solve the quadratic inequality by first finding the roots of the quadratic equation \(2x^2 - 5x + 2 = 0\). Use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=2\), \(b=-5\), and \(c=2\).
Once the roots are found, determine the intervals on the number line where the quadratic expression \(2x^2 - 5x + 2\) is greater than or equal to zero by testing values in each interval.
Write the domain of \(f(x)\) as the union of intervals where the inequality holds true, since these are the values of \(x\) for which the function is defined.

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:
3m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Domain of a Function

The domain of a function is the set of all input values (x-values) for which the function is defined. For functions involving square roots, the expression inside the root must be non-negative to produce real outputs. Identifying the domain ensures the function's outputs are real numbers.
Video consigliato:
3:51
Domain Restrictions of Composed Functions

Inequalities Involving Quadratic Expressions

To find where a quadratic expression is non-negative, solve the inequality by first finding the roots of the quadratic equation. Then, determine intervals where the quadratic is positive or zero by testing values or using the parabola's shape. This helps identify valid x-values for the function.
Video consigliato:
3:21
Nonlinear Inequalities

Factoring Quadratic Expressions

Factoring a quadratic expression involves rewriting it as a product of two binomials. This step is crucial to find the roots of the quadratic equation easily. For example, factoring 2x² - 5x + 2 helps locate critical points that define the domain boundaries.
Video consigliato:
06:08
Solving Quadratic Equations by Factoring
Pratica correlata
Domanda del libro di testo

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. x/(x + 2) ≥ 2

579
views
Domanda del libro di testo

Find the inverse of f(x) = x3 + 2

685
views
Domanda del libro di testo

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. r(x) = (x^2 + 4x + 3)/(x + 2)^2

693
views
Domanda del libro di testo

Among all pairs of numbers whose sum is 16, find a pair whose product is as large as possible. What is the maximum product?

1351
views
Domanda del libro di testo

Follow the seven steps to graph each rational function. f(x)=2x2/(x2−1)

32
views
Domanda del libro di testo

Find the domain of each function. f(x)=14x2−9x+2f(x) = \(\frac{1}{\sqrt{4x^2 - 9x + 2}\)}

521
views