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Understanding the domain and range of a logarithmic function is crucial when graphing transformations. The domain refers to all the possible input values (x-values) for which the function is defined. For the function \(f(x) = 4 - \log(x+6)\), we want the inside of the logarithm to be greater than 0, which means \(x + 6 > 0\) or \(x > -6\). Therefore, the domain is \[(-6, \infty)\].

The range, on the other hand, is the set of all possible output values (y-values). Since the basic log function \log(x)\ spans all real numbers in its range, and the transformations applied in the given function (reflection across the x-axis and vertical shift) do not change this property, the range consists of all real numbers, \((-\infty, \infty)\).

Horizontal shifts in logarithmic functions can significantly impact the shape and position of their graphs. A horizontal shift occurs when the input variable x is adjusted by adding or subtracting a constant. For the function \(f(x) = \log(x+6)\), the graph is shifted 6 units to the left due to the \(+6\) inside the log. This shift means that every point on the basic log function \(\log(x)\) is moved 6 units to the left along the x-axis.

Horizontal shifts do not change the range of the function but do alter its domain, which for this function moves from \((0, \infty)\) in the basic log graph to \[(-6, \infty)\]. This leftward shift also shifts the vertical asymptote from \(x=0\) to \(x=-6\).

Vertical shifts modify the graph of a function by moving it up or down along the y-axis. For the given function \(f(x) = 4 - \log(x+6)\), we observe a vertical shift because of the constant \(+4\) added to the term \(-\log(x+6)\). Rewriting the function as \(f(x) = -\log(x+6) + 4\) shows more clearly that it reflects the \log(x+6)\ function across the x-axis and then shifts it up by 4 units.

This vertical shift impacts the y-values of the function, moving every point on the graph upwards by 4 units. Consequently, the horizontal asymptote of the transformed function, originally at y=0 in the basic log function, moves up to y=4.

Graphing the function \(f(x) = 4 - \log(x+6)\) involves a few specific steps that incorporate all previously discussed transformations. Here's how you can sketch the graph step-by-step:

• Start with the basic log function \(\log(x)\).
• Apply the horizontal shift leftward by 6 units to get the graph of \(\log(x+6)\).
• Reflect the graph across the x-axis to obtain \(-\log(x+6)\).
• Finally, apply the vertical shift upward by 4 units, resulting in \(f(x) = -\log(x+6) + 4\).

This translated graph has a vertical asymptote at \(x = -6\) and a horizontal asymptote at \(y = 4\). As you plot points, keep these asymptotes in mind to ensure accuracy. By following these steps, you can visualize how the transformations combine to form the final graph.