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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.2.86
Chapter 7, Problem 7.2.86

86. Use a derivative to show that g(x)=√(x² + ln x) is one-to-one.

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Recall that a function is one-to-one if it is either strictly increasing or strictly decreasing on its domain. To use the derivative to show that \( g(x) = \sqrt{x^2 + \ln x} \) is one-to-one, we need to analyze the sign of \( g'(x) \).
First, identify the domain of \( g(x) \). Since \( \ln x \) is defined only for \( x > 0 \), the domain of \( g \) is \( (0, \infty) \).
Rewrite \( g(x) \) as \( g(x) = (x^2 + \ln x)^{1/2} \). Use the chain rule to find the derivative: \[ g'(x) = \frac{1}{2} (x^2 + \ln x)^{-1/2} \cdot \left( 2x + \frac{1}{x} \right) \].
Simplify the expression for \( g'(x) \): \[ g'(x) = \frac{2x + \frac{1}{x}}{2 \sqrt{x^2 + \ln x}} = \frac{2x + \frac{1}{x}}{2 \sqrt{x^2 + \ln x}}. \] Since the denominator \( 2 \sqrt{x^2 + \ln x} \) is positive for all \( x > 0 \), the sign of \( g'(x) \) depends on the numerator \( 2x + \frac{1}{x} \).
Analyze the numerator \( 2x + \frac{1}{x} \) for \( x > 0 \). Both terms are positive for all \( x > 0 \), so \( 2x + \frac{1}{x} > 0 \). Therefore, \( g'(x) > 0 \) for all \( x > 0 \), which means \( g(x) \) is strictly increasing on its domain and hence one-to-one.

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Key Concepts

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

Derivative and Monotonicity

The derivative of a function indicates its rate of change. If the derivative is always positive or always negative on an interval, the function is strictly increasing or decreasing there, respectively. This monotonic behavior implies the function is one-to-one on that interval.
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One-to-One Function (Injectivity)

A function is one-to-one if each output corresponds to exactly one input. This means no two different inputs produce the same output. Showing a function is strictly monotonic via its derivative is a common method to prove injectivity.
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Differentiation of Composite and Logarithmic Functions

To find the derivative of g(x) = √(x² + ln x), apply the chain rule and differentiate each component. The derivative of ln x is 1/x, and the derivative of x² is 2x. Combining these correctly is essential to analyze the sign of g'(x).
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