Let's work through the problem step-by-step.

Part (a): Sketch the Graph Using Transformations

1. Identify the Parent Function:
   The parent function is $f(x) = \log_3(x)$.

   The basic characteristics of $f(x) = \log_3(x)$ are:

   • Asymptote: $x = 0$
   • Domain: $x > 0$
   • Range: $(-\infty, \infty)$
   • A few key points are $(1, 0)$, $(3, 1)$, $\left( \frac{1}{3}, -1 \right)$.

2. Transformations of the Function $f(x) = \log_3(x+3) - 4$:

   • Horizontal Shift: The $x + 3$ inside the logarithm shifts the graph left by 3 units.
   • Vertical Shift: The $-4$ outside the logarithm shifts the graph down by 4 units.

3. Applying the Transformations:

   • New Asymptote: The horizontal shift of 3 units to the left moves the asymptote from $x = 0$ to $x = -3$. So, the asymptote for this function is: $x = -3$

   • Plot Key Points:

     • Start with the points of the parent function and apply the transformations to them:
       • Original Point (1, 0): Shift left by 3 units and down by 4 units to get $(-2, -4)$.
       • Original Point (3, 1): Shift left by 3 units and down by 4 units to get $(0, -3)$.
       • Original Point $\left( \frac{1}{3}, -1 \right)$: Shift left by 3 units and down by 4 units to get $\left( -\frac{8}{3}, -5 \right)$.
       • Choose one additional point, such as:
         • Original Point $\left(9, 2 \right)$: Shift left by 3 units and down by 4 units to get $\left(6, -2\right)$.

4. Graph Sketch:

   • Plot these transformed points: $(-2, -4)$, $(0, -3)$, $\left( -\frac{8}{3}, -5 \right)$, and $(6, -2)$.
   • Draw the vertical asymptote at $x = -3$.
   • Use these points to sketch the curve approaching the asymptote at $x = -3$ from the right, and moving upwards slowly as $x$ increases.

Part (b): Domain of $f(x)$

The domain is determined by the argument of the logarithmic function being positive: $x + 3 > 0$ $x > -3$ So, the domain of $f(x)$ is: $(-3, \infty)$

Part (c): Range of $f(x)$

Since this function involves a vertical shift of the logarithmic function, which has an original range of $(-\infty, \infty)$, the range remains: $(-\infty, \infty)$

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Would you like more details on any of these steps, or have any questions? Here are some related questions that might interest you:

1. What would happen if the function were $f(x) = \log_3(x + 3) + 4$?
2. How would the transformations change if we used a base 10 logarithm instead?
3. Can you determine the inverse function of $f(x) = \log_3(x + 3) - 4$?
4. How would a horizontal compression affect the graph of $f(x) = \log_3(x)$?
5. What is the effect of changing the base from 3 to another number?

Tip: When graphing transformations, applying shifts systematically (horizontal first, then vertical) helps to avoid errors.