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Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.
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Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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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 7
Find the domain of each rational function. f(x)=(x+7)/(x2+49)
Guida verificata passo dopo passo1
Identify the rational function given: \(f(x) = \frac{x+7}{x^{2} + 49}\).
Recall that the domain of a rational function includes all real numbers except where the denominator is zero, because division by zero is undefined.
Set the denominator equal to zero to find values to exclude: \(x^{2} + 49 = 0\).
Solve the equation \(x^{2} + 49 = 0\) by isolating \(x^{2}\): \(x^{2} = -49\).
Since \(x^{2} = -49\) has no real solutions (because a square cannot be negative in the real number system), the denominator is never zero for any real \(x\), so the domain is all real numbers.
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Domain of a Function
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions, the domain excludes any x-values that make the denominator zero, as division by zero is undefined.
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Domain Restrictions of Composed Functions
Rational Functions
A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding the behavior of the numerator and denominator polynomials is essential, especially identifying values that cause the denominator to be zero.
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6:04Intro to Rational Functions
Solving Quadratic Equations
To find values that make the denominator zero, you solve the quadratic equation Q(x) = 0. Techniques include factoring, completing the square, or using the quadratic formula to determine if real roots exist that restrict the domain.
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06:08Solving Quadratic Equations by Factoring
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