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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 3
Chapter 4, Problem 3

Find the domain of each rational function. g(x)=3x2/(x−5)(x+4)

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Identify the rational function given: \(g(x) = \frac{3x^{2}}{(x - 5)(x + 4)}\).
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 that are not in the domain: \((x - 5)(x + 4) = 0\).
Solve each factor for zero: \(x - 5 = 0\) gives \(x = 5\), and \(x + 4 = 0\) gives \(x = -4\).
Conclude that the domain of \(g(x)\) is all real numbers except \(x = 5\) and \(x = -4\).

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Key Concepts

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

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding rational functions involves recognizing that the function is undefined where the denominator Q(x) equals zero, which affects the domain.
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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, since division by zero is undefined.
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Domain Restrictions of Composed Functions

Finding Zeros of the Denominator

To find the domain of a rational function, identify the values of x that make the denominator zero by solving Q(x) = 0. These values are excluded from the domain because they cause the function to be undefined.
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Related Practice
Textbook Question

Determine which functions are polynomial functions. For those that are, identify the degree. g(x)=7x5−πx3+15xg(x)=7x^5-\(\pi\) x^3+\(\frac\)15x

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Textbook Question

In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=−x2+2x+3f(x) = -x^2 +2x + 3

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Textbook Question

In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x) = (x + 4)^2 - 2

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Textbook Question

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

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

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


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Textbook Question

Use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=3x4−11x3−x2+19x+6

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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 inversely as x. y = 12 when x = 5. Find y when x = 2.