What are the domain and range of [tex]f(x)=\log (x-1)+2[/tex]?
A. domain: [tex]x>1[/tex]; range: [tex]y>2[/tex]
B. domain: [tex]x>1[/tex]; range: all real numbers
C. domain: all real numbers; range: [tex]y>1[/tex]
D. domain: all real numbers; range: all real numbers
Answer (2)
The domain of f ( x ) = lo g ( x − 1 ) + 2 is determined by the inequality 0"> x − 1 > 0 .
Solving the inequality gives 1"> x > 1 , so the domain is 1"> x > 1 .
The range of the logarithmic function lo g ( x − 1 ) is all real numbers.
Adding 2 to the function does not change the range, so the range of f ( x ) is all real numbers. The final answer is domain: 1"> x > 1 ; range: all real numbers.
Explanation
Understanding the Function We are given the function f ( x ) = lo g ( x − 1 ) + 2 and asked to find its domain and range. Let's break this down.
Finding the Domain The domain of a logarithmic function is the set of all real numbers for which the argument of the logarithm is positive. In this case, the argument is x − 1 . So, we need to find all x such that 0"> x − 1 > 0 .
Determining the Domain Solving the inequality 0"> x − 1 > 0 , we add 1 to both sides to get 1"> x > 1 . Therefore, the domain of the function is all real numbers greater than 1.
Finding the Range Now let's find the range. The basic logarithmic function, y = lo g ( x ) , can take any real number as its output. The function f ( x ) = lo g ( x − 1 ) also has a range of all real numbers because x − 1 can take any positive value as x varies within the domain 1"> x > 1 .
Determining the Range Finally, we have f ( x ) = lo g ( x − 1 ) + 2 . Adding 2 to the logarithm shifts the graph vertically but does not change the range. Since lo g ( x − 1 ) can take any real value, lo g ( x − 1 ) + 2 can also take any real value. Thus, the range of the function is all real numbers.
Final Answer In conclusion, the domain of f ( x ) = lo g ( x − 1 ) + 2 is 1"> x > 1 , and the range is all real numbers.
Examples
Logarithmic functions are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, modeling population growth, and calculating the pH of a solution. Understanding the domain and range of logarithmic functions helps us to interpret these models correctly. For example, if we are modeling the population growth of a species using a logarithmic function, knowing the domain tells us the valid range of time for which the model is applicable, and the range tells us the possible population sizes.
The domain of the function f ( x ) = lo g ( x − 1 ) + 2 is 1"> x > 1 , and the range is all real numbers. Thus, the correct choice is A. domain: 1"> x > 1 ; range: all real numbers.
;
