For f(x) = log3 (x+3)-4   a) sketch the graph using transformations. Label any asymptotes with their equations, start by naming and stating the parent function, show at least 4 points on each graph.   b) Find the domain of f(x). c) find the range of f(x).

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:
\[
(-\infty, \infty)
\]

---

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.

Learn how to graph the function f(x) = log3(x + 3) - 4 using transformations. This solution explains how to find the domain and range, as well as identifying key transformations like horizontal and vertical shifts. Ideal for students in Grades 10-12 studying logarithmic functions.

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