Problem 73 Graph and state the domain and t... [FREE SOLUTION]

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Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 9: Problem 73

Graph and state the domain and the range of each function. $$ g(x)=\ln (x-1) $$

Short Answer

Expert verified

Domain: $ (1, \infty) $. Range: $ (-\infty, \infty) $.

Step by step solution

- Identify the Domain

To determine the domain of the function $ g(x) = \ln(x-1) $, we need to identify the values of $ x $ for which the function is defined. Since the natural logarithm function $ \ln(y) $ is only defined for $ y > 0 $, we need $ x-1 > 0 $. Solving this inequality, we get $ x > 1 $. Thus, the domain is $ (1, \infty) $.

- Identify the Range

The range of the natural logarithm function $ \ln(y) $ is all real numbers $ (-\infty, \infty) $. Since there are no restrictions on the output of the natural logarithm, the range of $ g(x) = \ln(x-1) $ is also $ (-\infty, \infty) $.

- Graph the Function

To graph $ g(x) = \ln(x-1) $, start by shifting the graph of $ y = \ln(x) $ to the right by 1 unit. The vertical asymptote, originally at $ x = 0 $ for $ y = \ln(x) $, will shift to $ x = 1 $. The graph will be defined for all $ x > 1 $ and will increase slowly to the right while approaching negative infinity as $ x $ approaches 1 from the right.

- Verify the Function Behavior

Check points to verify: For $ x = 2 $, $ g(2) = \ln(2-1) = \ln(1) = 0 $. For $ x = 3 $, $ g(3) = \ln(3-1) = \ln(2) $, which is approximately 0.693. These points ensure the accuracy of the graph.

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Key Concepts

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

natural logarithm

The natural logarithm, represented as $\ln(x)$, is a logarithmic function with the base $e e$, where $e$ is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm of a number is the power to which $e$ must be raised to obtain that number. Importantly, $\ln(x)$ is only defined for positive values of $x$, meaning $x > 0$.Properties of the Natural Logarithm:

$\ln(1) = 0$ because $e^0 = 1$.
$\ln(e) = 1$ because $e^1 = e$.
The natural logarithm function is continuous and increases without bound, but at a decreasing rate.

In the function $g(x) = \ln(x-1)$, the argument $x-1$must be greater than zero. Consequently, the domain is $(1, \infty)$.

graphing functions

Graphing functions helps us visualize their behavior and characteristics. For the function $g(x) = \ln(x-1)$, we can follow these steps to create its graph:Step-by-Step Graphing Guide:

Identify the base function: Start with the graph of $y = \ln(x)$. This curve passes through the point $(1, 0)$ and approaches negative infinity as $x$ approaches zero from the right.
Shift the graph horizontally: Since our function is $g(x) = \ln(x-1)$, we shift the entire graph of $y = \ln(x)$ one unit to the right. This places the vertical asymptote at $x = 1$.
Plot key points: Verify specific points on the graph. For instance, when $x = 2$, $g(2) = \ln(1) = 0$. Likewise, when $x = 3$, $g(3) = \ln(2) \approx 0.693$.

These steps ensure a precise and accurate representation of the function on the graph.

vertical asymptote

A vertical asymptote is a line that a graph approaches but never touches or crosses. It represents values where the function grows infinitely large or small. In the context of $g(x) = \ln(x-1)$, the vertical asymptote occurs at $x = 1$.Understanding Vertical Asymptotes:

Vertical asymptotes are found by determining the values for which the function is undefined or tends towards infinity.
For natural logarithm functions, such as $g(x) = \ln(x-1)$, the vertical asymptote is determined by setting the argument of the logarithm equal to zero. So, $x-1 = 0$ implies $x = 1$.

The graph of $g(x) = \ln(x-1)$ approaches the line $x = 1$ from the right but never crosses it. As $x$ gets closer to 1, $g(x)$ decreases without bound, creating a distinct vertical boundary on the graph.

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91影视

Graph and state the domain and the range of each function. $$ g(x)=\ln (x-1) $$

Short Answer

Step by step solution

- Identify the Domain

- Identify the Range

- Graph the Function

- Verify the Function Behavior

Key Concepts

natural logarithm

graphing functions

vertical asymptote

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