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)
The domain of f ( x ) = lo g ( x + 6 ) − 4 is determined by the inequality 0"> x + 6 > 0 , which simplifies to -6"> x > − 6 .
The range of a logarithmic function is all real numbers.
Subtracting a constant from the logarithmic function does not change the range.
Therefore, the domain is -6"> x > − 6 and the range is all real numbers, so the answer is -6; range: all real numbers}"> d o main : x > − 6 ; r an g e : a ll re a l n u mb ers .
Explanation
Understanding the Problem We are asked to find the domain and range of the function f ( x ) = lo g ( x + 6 ) − 4 . Let's break this down.
Finding the Domain The domain of a logarithmic function is all values of x for which the argument of the logarithm is positive. In this case, the argument is x + 6 . So we need to solve the inequality 0"> x + 6 > 0 . Subtracting 6 from both sides, we get -6"> x > − 6 . Therefore, the domain is all real numbers greater than -6.
Finding the Range The range of a logarithmic function is all real numbers. The function lo g ( x + 6 ) can take any real value. Subtracting 4 from the logarithm shifts the graph vertically but does not change the range. Therefore, the range of f ( x ) = lo g ( x + 6 ) − 4 is all real numbers.
Final Answer Therefore, the domain of the function 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. Understanding the domain and range of logarithmic functions helps us to interpret these models and make accurate predictions. For example, in seismology, the Richter scale uses logarithms to quantify the magnitude of earthquakes. The formula is M = lo g 10 ( A ) − lo g 10 ( A 0 ) , where A is the amplitude of the seismic waves and A 0 is a reference amplitude. The domain of this function is 0"> A > 0 , since the amplitude must be positive. The range is all real numbers, as the magnitude can be any real number depending on the amplitude.
The domain of the function f ( x ) = lo g ( x + 6 ) − 4 is -6"> x > − 6 and the range is all real numbers. Thus, the answer is option B. This means the function can accept all inputs greater than -6 and can output any real number.
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