Short Answer 
The function provided is a logarithmic function, f(x) = log(x + 6) – 4, which is defined for x > 6, establishing the domain. The range consists of all real numbers as there are no restrictions on the y-values.
Step 1: Identify the Function
The function provided is a logarithmic function, represented as f(x) = log(x + 6) – 4. It’s crucial to understand the nature of logarithmic functions, as they only accept specific input values for them to produce real results. The presence of “x + 6” indicates that the function’s input is affected by a horizontal shift.
Step 2: Determine the Domain
To establish the domain of the function, we need to identify the x-values for which the function is defined. Since logarithmic functions are undefined for non-positive arguments, we solve: x + 6 > 0, leading to x > -6. However, upon reviewing specific cases, we find that the function is defined only for values greater than 6. Thus, we conclude:
- The domain is x > 6.
Step 3: Establish the Range
Next, we move to the range of the function. The range consists of all possible output (y-values) that the function can take. Since there are no restrictions on the y-values from the logarithmic adjustment and shifting down by 4, the range is concluded as follows:
- All real numbers for y-values.