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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 7
Chapter 4, Problem 7

Find the domain of each rational function. f(x)=(x+7)/(x2+49)

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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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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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Related Practice
Textbook Question

The graph of a quadratic function is given. Write the function's equation, selecting from the following options.

f(x)=x2+2x+1g(x)=x2−2x+1f(x) = x^2 + 2x + 1 \(\quad\) g(x) = x^2 - 2x + 1

h(x)=x2−1j(x)=−x2−1h(x) = x^2 - 1 \(\quad\) j(x) = -x^2 - 1

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Textbook Question

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. x2+5x+4>0

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Textbook Question

Divide using long division. State the quotient, and the remainder, r(x)r(x). (12x2+x−4)÷(3x−2)(12x^2+x-4)\(\div\)(3x-2)

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Textbook Question

Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies jointly as x and z. y = 25 when x = 2 and z = 5. Find y when x = 8 and z = 12.

Textbook Question

Determine which functions are polynomial functions. For those that are, identify the degree. f(x)=x12−3x2+5f(x)=x^{\(\frac\)12}-3x^2+5

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Textbook Question

In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree. f(x)=x1/3 −4x2+7