Math Problem Statement

Domain and range

Domain and range of the function f(x) = ln(x + 1)

Solution

The given function is:

f(x)=ln(x+1)f(x) = \ln(x + 1)

Domain:

The natural logarithm function, ln(y)\ln(y), is only defined when y>0y > 0. Therefore, for f(x)=ln(x+1)f(x) = \ln(x + 1) to be defined, we must have:

x+1>0x + 1 > 0

Solving this inequality:

x>1x > -1

Thus, the domain of the function is:

(1,)(-1, \infty)

Range:

The range of the natural logarithm function ln(y)\ln(y) is all real numbers, as it can take any value from -\infty to \infty. Since there are no restrictions placed by the function x+1x + 1, the range of f(x)f(x) is:

(,)(-\infty, \infty)

Summary:

  • Domain: (1,)(-1, \infty)
  • Range: (,)(-\infty, \infty)

Would you like further details on how the logarithmic function behaves or any additional clarifications?

Here are five related questions to expand this topic:

  1. What is the derivative of f(x)=ln(x+1)f(x) = \ln(x + 1)?
  2. How does shifting the logarithmic function affect its graph?
  3. Can you calculate the inverse of f(x)=ln(x+1)f(x) = \ln(x + 1)?
  4. What is the behavior of f(x)=ln(x+1)f(x) = \ln(x + 1) as x1x \to -1?
  5. How does the range of f(x)f(x) compare to other logarithmic functions, such as ln(x)\ln(x)?

Tip: When determining the domain of logarithmic functions, always ensure the argument inside the logarithm is positive.

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Math Problem Analysis

Mathematical Concepts

Logarithmic Functions
Domain and Range

Formulas

f(x) = ln(x + 1)

Theorems

Logarithmic function properties

Suitable Grade Level

Grades 10-12