What are the domain and range of [tex]f(x)=\log (x+6)-4[/tex]?
A. domain: [tex]x>-6[/tex]; range: [tex]y>4[/tex]
B. domain: [tex]x>-6[/tex]; range: all real numbers
C. domain: [tex]x>6[/tex]; range: [tex]y>-4[/tex]
D. domain: [tex]x>6[/tex]; range: all real numbers
Answer (2)
Determine the domain by solving the inequality 0"> x + 6 > 0 , which gives -6"> x > − 6 .
The domain is all real numbers x such that -6"> x > − 6 .
The range of the logarithmic function is all real numbers.
The domain is -6"> x > − 6 and the range is all real numbers, so the answer is -6; \text{ range: all real numbers}}"> domain: x > − 6 ; range: all real numbers
Explanation
Understanding the Function We are given the function f ( x ) = lo g ( x + 6 ) − 4 and we need to find its domain and range.
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 + 6 . So, we need to find all x such that 0"> x + 6 > 0 .
Determining the Domain Solving the inequality 0"> x + 6 > 0 , we subtract 6 from both sides to get -6"> x > − 6 . Therefore, the domain of the function is all real numbers greater than -6.
Finding the Range The range of a logarithmic function is the set of all real numbers. The basic logarithmic function lo g ( x ) can take any real value. Since f ( x ) is a transformation of the basic logarithmic function, specifically a horizontal shift by 6 units to the left and a vertical shift by 4 units down, the range remains all real numbers.
Conclusion Therefore, the domain of f ( x ) = lo g ( x + 6 ) − 4 is -6"> x > − 6 , 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 in chemistry. Understanding the domain and range of logarithmic functions helps us to interpret these models correctly and make accurate predictions. For example, in earthquake intensity measurement, the magnitude R is given by R = lo g ( A 0 A ) , where A is the amplitude of the seismic waves and A 0 is a reference amplitude. The argument of the logarithm must be positive, which means that A must be greater than 0. Similarly, in population growth models, the population size at time t can be modeled using exponential and logarithmic functions, and the domain and range of these functions determine the possible values of time and population size.
The domain of the function f ( x ) = lo g ( x + 6 ) − 4 is all real numbers -6"> x > − 6 , and the range is all real numbers. Therefore, the correct option is A. domain: -6"> x > − 6 ; range: all real numbers.
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