Graphing Logarithmic Function f(x) = ln(x + 1) with Domain and Range

Math Problem Statement

Use your graphing calculator to produce a graph of the function. Then determine the domain and range of the function by looking at its graph. f left parenthesis x right parenthesis equals ln left parenthesis x plus 1 right parenthesis Question content area bottom Part 1 Identify the graph of the function f left parenthesis x right parenthesis equals ln left parenthesis x plus 1 right parenthesis. Choose the correct graph below. A.

A coordinate system has a horizontal axis labeled from negative 6 to 6 in increments of 1 and a vertical axis labeled from negative 6 to 6 in increments of 1. A curve rises steeply to the left of the vertical axis and then rises at a decreasing rate, passing through the points (5, 1.8) and (6, 1.9). As the curve approaches a horizontal cooridnate of negative 1 from the right, the curve approaches negative infinity. All coordinates are approximate. [negative 6,6] by [minus6,6] B.

A coordinate system has a horizontal axis labeled from negative 2 to 10 in increments of 1 and a vertical axis labeled from negative 6 to 6 in increments of 1. A curve rises steeply to the right of the vertical axis and then rises at a decreasing rate, passing through (9, 2.1) and (10, 2.2). As the curve approaches a horizontal coordinate of 1 from the right, the curve approaches negative infinity. All coordinates are approximate. [negative 2,10] by [minus6,6] C.

A coordinate system has a horizontal axis labeled from negative 6 to 6 in increments of 1 and a vertical axis labeled from negative 6 to 6 in increments of 1. A curve falls steeply to the left of the vertical axis and then falls at a decreasing rate, passing through the points (5, negative 1.8) and (6, negative 1.9). As the curve approaches a horizontal coordinate of negative 1 from the right, the curve approaches positive infinity. All coordinates are approximate. [negative 6,6] by [minus6,6] D.

A coordinate system has a horizontal axis labeled from negative 2 to 10 in increments of 1 and a vertical axis labeled from negative 6 to 6 in increments of 1. A curve falls steeply to the right of the vertical axis and then falls at a decreasing rate, passing through (9, negative 2.1) and (10, negative 2.2). As the curve approaches a horizontal coordinate of 1 from the right, the curve approaches positive infinity. All coordinates are approximate. [negative 2,10] by [minus6,6] Part 2 The domain of the function is enter your response here. (Type your answer in interval notation.) Part 3 The range of the function is enter your response here. (Type your answer in interval notation.)

Solution

Part 1: Identify the graph of the function $f(x) = \ln(x + 1)$

The function $f(x) = \ln(x + 1)$ is a natural logarithmic function shifted left by 1 unit. It has the following key features:

The logarithmic function $\ln(x)$ has a vertical asymptote at $x = 0$ . For $f(x) = \ln(x + 1)$ , the vertical asymptote shifts left by 1 unit to $x = -1$ .
The function is undefined for $x \leq -1$ because the natural logarithm function is only defined for positive arguments.
The graph approaches negative infinity as $x$ approaches $-1$ from the right.
As $x \to \infty$ , $f(x) \to \infty$ , but the rate of increase slows down.

The correct graph will show a vertical asymptote at $x = -1$ , with the curve increasing to the right of this asymptote. Based on this analysis:

Graph A best fits the description of $f(x) = \ln(x + 1)$ . The curve passes through points like (5, 1.8) and (6, 1.9) and rises at a decreasing rate while having a vertical asymptote at $x = -1$ .

Thus, the correct graph is Graph A.

Part 2: Domain of the function $f(x) = \ln(x + 1)$

For the function $f(x) = \ln(x + 1)$ , the argument $x + 1$ must be greater than 0 for the logarithm to be defined: $x + 1 > 0 \implies x > -1$ Thus, the domain of the function is $(-1, \infty)$ .

Domain: $(-1, \infty)$

Part 3: Range of the function $f(x) = \ln(x + 1)$

The natural logarithmic function $\ln(x + 1)$ can take any real value as $x \to -1$ from the right and approaches negative infinity, and as $x \to \infty$ , the function increases without bound. Therefore, the range of the function is all real numbers, $(-\infty, \infty)$ .

Range: $(-\infty, \infty)$

Do you have any questions or would you like more details?

Here are 5 related questions to deepen understanding:

What is the effect of adding or subtracting a constant inside the logarithm function on its graph?
How would the graph of $f(x) = \ln(x + 2)$ differ from $f(x) = \ln(x + 1)$ ?
How do you find the vertical asymptote of a logarithmic function?
What happens to the domain and range if we reflect the function $\ln(x + 1)$ across the x-axis?
What is the inverse function of $f(x) = \ln(x + 1)$ ?

Tip: Always remember that the natural logarithm function $\ln(x)$ is undefined for non-positive values of $x$ , which directly affects the domain of the function.

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Math Problem Analysis

Mathematical Concepts

Logarithmic Functions
Domain and Range
Graphing

Formulas

f(x) = ln(x + 1)

Theorems

Properties of Logarithmic Functions
Domain and Range of Logarithmic Functions

Suitable Grade Level

Grades 10-12

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