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Determine the transformations that produce the graph of the functions g(x) = -3log(x - 8) - 9 and h(x) = -1/3 log(x - 8) + 9 from the parent function f(x) = log(x). Then compare the similarities and differences between the two functions, including the domain and range.

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Final Answer:

The transformations applied to the parent function
\(f(x) = \log(x)\) to obtain
\(g(x) = -3\log(x - 8) - 9\) involve a vertical compression by a factor of \(1/3\), a horizontal shift to the right by 8 units, a vertical reflection, and a vertical shift downward by 9 units. Similarly, for
\(h(x) = -1/3 \log(x - 8) + 9\), the transformations include a vertical stretch by a factor of 3, a horizontal shift to the right by 8 units, and a vertical shift upward by 9 units. The differences lie in the vertical stretches or compressions and the direction of the shifts.

Step-by-step explanation:

To obtain
\(g(x)\), the coefficient
\(-3\) indicates a vertical compression by a factor of \(1/3\) compared to the parent function
\(f(x)\). The horizontal shift of 8 units to the right is denoted by
\((x - 8)\), and the negative sign before the logarithmic term reflects the graph vertically. The final vertical shift downward by 9 units is represented by
\(-9\) outside the logarithmic term.

For
\(h(x)\), the coefficient
\(-1/3\) implies a vertical stretch by a factor of 3. The horizontal shift to the right by 8 units is indicated by
\((x - 8)\), and the positive sign before the logarithmic term reflects the graph vertically. The final vertical shift upward by 9 units is represented by
\(+9\) outside the logarithmic term.

The comparison between the two functions reveals that
\(g(x)\) has a vertical compression and downward shift, while
\(h(x)\) has a vertical stretch and an upward shift. These transformations affect the domain and range differently, altering the scale and position of the logarithmic graph.

User DHamrick
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