Q. 134 Is there any function of the for... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 5: Q. 134 (page 301)

Is there any function of the form $y = x^{a}, 0 < 伪 < 1,$ that increases more slowly than a logarithmic function whose base is greater than 1? Explain.

Short Answer

Expert verified

There is no function in the form $y = x^{a}, 0 < 伪 < 1,$

Step by step solution

Step 1. Introduction

An exponential function (with a base greater than $1)$ grows faster than a polynomial form $f (x) = x^{n}, n 鈭� \{1, 2, 3, . . .\} .$ We need to explain is there any function of the form $y = x^{a}, 0 < a < 1$ that increases more slowly than a logarithmic function whose base is greater than $1 .$

Step 2. Explanation

Since the exponential function is inverse to the logarithmic function,

$f (x) = a^{x} 鈬� f^{- 1} (x) = \log_{a} x, a > 1,$ and the polynomial is the inverse of the root $f (x) = x^{n}, n 鈭� \{1, 2, . . .\} .$

$f^{- 1} (x) = \sqrt[n]{x} = x^{\frac{1}{n}} = x^{伪}, 伪 = \frac{1}{n}, 0 < 伪 < 1,$ then the properties are inherited to the inverse function. That is we conclude that there is no function of the form $y = x^{伪}, 0 < 伪 < 1,$ that increases more slowly than a logarithmic whose base is greater than $1 .$

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91影视

Is there any function of the form $y = x^{a}, 0 < 伪 < 1,$ that increases more slowly than a logarithmic function whose base is greater than 1? Explain.

Short Answer

Step by step solution

Step 1. Introduction

Step 2. Explanation

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91影视

Is there any function of the form y=xa,0<伪<1,that increases more slowly than a logarithmic function whose base is greater than 1? Explain.

Short Answer

Step by step solution

Step 1. Introduction

Step 2. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Pure Maths

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Study anywhere. Anytime. Across all devices.

Is there any function of the form $y = x^{a}, 0 < 伪 < 1,$ that increases more slowly than a logarithmic function whose base is greater than 1? Explain.