For the function $y=\ln (x+1)+5$, which of the following statements is true?
A. The domain is $(1, \infty)$, and the range is all real numbers.
B. The domain is $(-1, \infty)$, and the range is all real numbers.
C. The domain is all real numbers, and the range is $[5, \infty)$.
D. The domain is $(1, \infty)$, and the range is $[5, \infty)$.
Answer (2)
Determine the domain of the function y = ln ( x + 1 ) + 5 by solving the inequality 0"> x + 1 > 0 , which gives -1"> x > − 1 .
The domain is ( − 1 , ∞ ) .
The range of the natural logarithm function is all real numbers, and adding a constant does not change the range.
The range of the function is all real numbers. Therefore, the answer is B .
Explanation
Problem Analysis We are given the function y = ln ( x + 1 ) + 5 and need to find its domain and range.
Finding the Domain The domain of the natural logarithm function ln ( u ) is all 0"> u > 0 . Therefore, for our function, we must have 0"> x + 1 > 0 . Solving this inequality gives us -1"> x > − 1 . Thus, the domain of the function is ( − 1 , ∞ ) .
Finding the Range The range of the natural logarithm function ln ( u ) is all real numbers. Adding a constant to the logarithm function shifts the graph vertically but does not change the range. Therefore, the range of y = ln ( x + 1 ) + 5 is also all real numbers.
Selecting the Correct Option Comparing our findings with the given options, we see that option B matches our results: The domain is ( − 1 , ∞ ) , and the range is all real numbers.
Final Answer Therefore, the correct answer is B.
Examples
Understanding the domain and range of logarithmic functions is crucial in many real-world applications. For example, in finance, the growth of an investment can sometimes be modeled using logarithmic scales. Knowing the domain helps determine the valid input values (e.g., time or initial investment), while the range helps understand the possible outcomes (e.g., the final value of the investment). Similarly, in physics, logarithmic scales are used to measure the intensity of earthquakes (Richter scale) or the loudness of sound (decibels), where understanding the domain and range is essential for interpreting the measurements correctly. Logarithmic functions are also used in computer science to analyze the efficiency of algorithms, where the domain represents the input size and the range represents the time or space complexity.
The domain of the function y = ln ( x + 1 ) + 5 is ( − 1 , ∞ ) , and the range is all real numbers. Therefore, the correct answer is B.
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