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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 5
Chapter 4, Problem 5

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

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Identify the rational function given: \(h(x) = \frac{x+7}{x^{2} - 49}\).
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 the values to exclude: \(x^{2} - 49 = 0\).
Solve the equation \(x^{2} - 49 = 0\) by factoring it as a difference of squares: \((x - 7)(x + 7) = 0\).
Find the roots from the factors: \(x - 7 = 0\) gives \(x = 7\), and \(x + 7 = 0\) gives \(x = -7\). These values are excluded from the domain.

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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 input values (x-values) for which the function is defined. For rational functions, the domain excludes 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 how to simplify and analyze these functions is essential, especially identifying values that cause the denominator Q(x) to be zero.
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Factoring Polynomials

Factoring involves rewriting polynomials as products of simpler polynomials. For example, x² - 49 factors as (x - 7)(x + 7). Factoring helps identify zeros of the denominator to determine excluded values from the domain.
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Introduction to Factoring Polynomials
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

Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies directly as x and inversely as the square of z. y = 20 when x = 50 and z = 5. Find y when x = 3 and z = 6.

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Textbook Question

Determine which functions are polynomial functions. For those that are, identify the degree. h(x)=7x3+2x2+1xh(x)=7x^3+2x^2+\(\frac{1}{x}\)

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Textbook Question

Divide using long division. State the quotient, and the remainder, r(x)r(x). (6x3+7x2+12x−5)÷(3x−1)(6x^3+7x^2+12x-5)\(\div\)(3x-1)

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Textbook Question

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

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Textbook Question

In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=3x4−11x3−3x2−6x+8

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