What are the domain and range of $f(x)=\log (x+6)-4$?
A. domain: $x>-6$; range: $y>4$
B. domain: $x>-6$; range: all real numbers
C. domain: $x>6$; range: $y>-4$
D. domain: $x>6$; range: all real numbers
Answer (1)
The domain of the function f ( x ) = lo g ( x + 6 ) − 4 is determined by the argument of the logarithm being positive: 0"> x + 6 > 0 .
Solving the inequality 0"> x + 6 > 0 gives -6"> x > − 6 .
The range of the standard logarithmic function lo g ( x ) is all real numbers, and subtracting a constant does not change this.
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 want to find the domain and range of the function f ( x ) = lo g ( x + 6 ) − 4 . The domain is the set of all possible input values (x-values) for which the function is defined. The range is the set of all possible output values (y-values) that the function can take.
Finding the Domain The logarithm function, lo g ( x ) , is only defined for positive values of x . Therefore, for f ( x ) = lo g ( x + 6 ) − 4 to be defined, we must have 0"> x + 6 > 0 . Solving this inequality for x , we get:
0"> x + 6 > 0 -6"> x > − 6
So, the domain of the function is all real numbers x such that -6"> x > − 6 .
Finding the Range The range of the standard logarithmic function lo g ( x ) is all real numbers. In our case, we have lo g ( x + 6 ) , where x + 6 can take any positive value as x varies over the domain -6"> x > − 6 . Therefore, lo g ( x + 6 ) can take any real value.
Now, we have f ( x ) = lo g ( x + 6 ) − 4 . Since lo g ( x + 6 ) can take any real value, subtracting 4 from it will not change the range. Thus, the range of f ( x ) is also all real numbers.
Final Answer 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 determining the pH of a solution in chemistry. Understanding the domain and range of logarithmic functions helps us to interpret and apply these models correctly. For example, when modeling population growth with a logarithmic function, the domain tells us the valid range of time values for which the model is applicable, and the range tells us the possible population sizes that the model can predict.
