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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 1
Chapter 4, Problem 1

Find the domain of each rational function. f(x)=5x/(x−4)

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1
Identify the rational function given: \(f(x) = \frac{5x}{x - 4}\).
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 to exclude: \(x - 4 = 0\).
Solve the equation for \(x\): \(x = 4\).
Conclude that the domain of \(f(x)\) is all real numbers except \(x = 4\), which can be written in interval notation as \((-\infty, 4) \cup (4, \infty)\).

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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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Intro to Rational Functions

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 Values that Make the Denominator Zero

To find the domain of a rational function, set the denominator equal to zero and solve for x. These values are excluded from the domain because they cause division by zero, which is undefined in mathematics.
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Related Practice
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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. (x−4)(x+2)>0

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Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies directly as x. y = 65 when x = 5. Find y when x = 12.

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Determine which functions are polynomial functions. For those that are, identify the degree. f(x)=5x2+6x3f(x)=5x^2+6x^3

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Divide using long division. State the quotient, and the remainder, r(x). (x2+8x+15)÷(x+5)

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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

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+1)2+4 f(x) = - (x + 1)^2 + 4

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