Ch. 3 - Polynomial and Rational Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 1

Chapter 4, Problem 1

Provide a short answer to each question. What is the domain of the function ƒ(x)=1/x? What is its range?

Verified step by step guidance

Identify the function given: $f(x) = \frac{1}{x}$. This is a rational function where the denominator cannot be zero.

Determine the domain by finding all values of $x$ for which the function is defined. Since division by zero is undefined, exclude $x = 0$ from the domain.

Express the domain in set notation: all real numbers except zero, which can be written as $\{ x \in \mathbb{R} \mid x \neq 0 \}$.

To find the range, consider the possible output values of $f(x)$. Since $f(x) = \frac{1}{x}$, as $x$ approaches zero from either side, $f(x)$ grows without bound positively or negatively, and as $x$ becomes very large or very small, $f(x)$ approaches zero but never equals zero.

Conclude that the range is all real numbers except zero, expressed as $\{ y \in \mathbb{R} \mid y \neq 0 \}$.

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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 ƒ(x) = 1/x, the function is undefined when the denominator is zero, so x cannot be zero. Therefore, the domain includes all real numbers except zero.

Range of a Function

The range of a function is the set of all possible output values (y-values) that the function can produce. For ƒ(x) = 1/x, the output can be any real number except zero, because 1/x never equals zero for any real x. Thus, the range is all real numbers except zero.

Rational Functions and Undefined Points

A rational function is a ratio of two polynomials, and it is undefined where the denominator equals zero. In ƒ(x) = 1/x, the denominator x cannot be zero, which creates a vertical asymptote at x = 0. Understanding this helps identify domain restrictions and behavior near undefined points.

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Related Practice

Textbook Question

Provide a short answer to each question. What is the domain of the function $ƒ (x) = \frac{1}{x^{2}}$ ? What is its range?

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Textbook Question

Graph each quadratic function. Give the vertex, axis, x-intercepts, y-intercept, domain, range, and largest open intervals of the domain over which each function is increasing or decreasing. ƒ(x)=-3x²-12x-1

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Textbook Question

Fill in the blank(s) to correctly complete each sentence. A polynomial function with leading term 3x⁵ has degree ____.

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Determine whether each statement is true or false. If false, explain why. Because x-1 is a factor of ƒ(x)=x⁶-x⁴+2x²-2, we can also conclude that ƒ(1) = 0.

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Textbook Question

Determine whether each statement is true or false. If false, explain why. For ƒ(x)=(x+2)⁴(x-3), the number 2 is a zero of multiplicity 4.

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