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Chapter 1: Problem 99

\(g(x)=\sqrt[3]{x}\) is an odd root function

Short Answer

Expert verified
Yes, \( g(x) = \sqrt[3]{x} \) is an odd function because \( g(-x) = -g(x) \).

Step by step solution

01

Understanding the Function

The function given is a cube root function, specifically, it is \( g(x) = \sqrt[3]{x} \). This function implies that for any input value \( x \), it gives the cube root of \( x \). Cube root functions are defined for all real numbers, including both positive and negative inputs.
02

Identifying Odd Functions

An odd function is characterized by the property \( f(-x) = -f(x) \) for all \( x \). This means that if you substitute \( -x \) into the function, the result should be the negative of what you get when substituting \( x \).
03

Verify the Odd Function Property

To verify if \( g(x) = \sqrt[3]{x} \) is an odd function, we compute \( g(-x) \). So, \( g(-x) = \sqrt[3]{-x} = -\sqrt[3]{x} = -g(x) \). Since \( g(-x) = -g(x) \), the function satisfies the condition for being odd.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Odd Function
An odd function, in mathematical terms, is a type of symmetry that functions can exhibit. It is defined by the equation \( f(-x) = -f(x) \) for every real number \( x \). This property can be thought of as rotational symmetry around the origin in a Cartesian plane. Simply put, if you reflect the graph of the function across the origin, it remains unchanged.
  • Example: If \( f(x) = x^3 \), then \( f(-x) = (-x)^3 = -x^3 = -f(x) \). This confirms that \( x^3 \) is an odd function.
  • Significance: Odd functions are especially useful in calculus and physics for having zero net area under the curve over symmetric intervals.
Understanding odd functions helps in recognizing patterns in equations and understanding their graphical representations. This concept is foundational in higher-level mathematics.
Real Numbers
Real numbers are a set of numbers that include all the numbers on the number line. They comprise both rational numbers (like fractions and integers) and irrational numbers (like \( \sqrt{2} \) and \( \pi \)).
Real numbers are vital when dealing with functions because they ensure that you can input and obtain smooth, continuous values without any gaps.
  • Characteristics: Real numbers can be positive, negative, or zero. They are commonly used in algebra, calculus, and real-world measurements.
  • Examples:
    • Rational Numbers: \( 1, \frac{1}{2}, 0.75 \).
    • Irrational Numbers: \( \sqrt{3}, \pi \).
In the context of the cube root function \( g(x) = \sqrt[3]{x} \), this means the function can handle any real number as input, producing a real number as output.
Function Properties
Functions, like \( g(x) = \sqrt[3]{x} \), have specific properties that define how they behave and are interpreted. Some common function properties include domain, range, odd and even nature, and continuity.
  • Domain: The set of all possible input values. For cube root functions, the domain is all real numbers since the cube root of a negative number is well-defined.
  • Range: The set of possible output values. For cube root functions, the range is also all real numbers, as any real number is the cube root of some other real number.
  • Continuity: A function is continuous if there are no breaks or jumps in its graph. Cube root functions are continuous across their domain.
Understanding these properties allows you to better analyze and predict the behavior of functions, making them a powerful tool in both theoretical and applied mathematics.

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Most popular questions from this chapter

Find the length of the arc intercepted by central angle \(\theta\) in a circle of radius \(r\) . Round to the nearest hundredth. a. \(r=12.8 \mathrm{cm}, \theta=\frac{5 \pi}{6} \mathrm{rad} \quad \) b. \(r=4.378 \mathrm{cm}, \theta=\frac{7 \pi}{6} \mathrm{rad} \quad\) c. \(r=0.964 \quad \mathrm{cm}, \quad \theta=50^{\circ} \quad\) d. \(r=8.55 \mathrm{cm},\) \(\theta=325^{\circ}\)

For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms. $$ \log x^{4} y $$

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[T] A condominium in an upscale part of the city was purchased for \(\$ 432,000\) . In 35 years it is worth \(\$ 60,500\) . Find the rate of depreciation.
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