https://file.aiquickdraw.com/mathbot/file/1726798702288172e5mme.jpg

Let's analyze the graph and answer each part of the problem.

### Part (a): Find \( f(-3) \).
From the graph, locate \( x = -3 \) on the x-axis and trace up or down to where the function intersects. It appears that \( f(-3) = 0 \).

### Part (b): When is \( f(x) = -1 \)?
To solve this, find where the function reaches \( y = -1 \). By tracing along the horizontal line at \( y = -1 \), it intersects the function at \( x = 2 \). So, \( f(x) = -1 \) when \( x = 2 \).

### Part (c): Find the domain of \( f(x) \).
The domain refers to all possible \( x \)-values for which the function is defined. The graph extends infinitely in both the positive and negative \( x \)-directions, so the domain of \( f(x) \) is:
\[
\text{Domain} = (-\infty, \infty)
\]

### Part (d): Find the range of \( f(x) \).
The range is the set of possible \( y \)-values. From the graph, we can see that the function approaches \( y = 0 \) from below as \( x \to \infty \), but it never crosses the x-axis. It also extends down to negative infinity. Thus, the range is:
\[
\text{Range} = (-\infty, 0)
\]

### Part (e): Is this function one-to-one?
A function is one-to-one if it passes the horizontal line test (no horizontal line intersects the graph more than once). Since every horizontal line intersects the graph only once, the function is indeed one-to-one.

### Part (f): Write the equation of \( f(x) \).
The graph looks like a transformed version of the natural logarithmic function \( \ln(x) \), shifted and reflected. Based on the shape and behavior, the function resembles \( f(x) = -\ln(x+4) \), where the graph is shifted left by 4 units.

Would you like more details on any of these parts? Here are some follow-up questions to expand understanding:

1. What is the general shape of a logarithmic function, and how does it behave at different values of \( x \)?
2. How can we determine if a function is continuous based on its graph?
3. What are the properties of a one-to-one function, and why is this important for finding an inverse function?
4. How does a transformation like \( -\ln(x+4) \) affect the graph of \( \ln(x) \)?
5. How can the domain and range of a logarithmic function be determined algebraically?

**Tip:** To write the equation of a graph, always look for transformations (shifts, stretches, reflections) applied to well-known parent functions like logarithmic or exponential functions.

Use the graph of the function f(x) to answer the following: (a) Find f(-3). (b) When is f(x) = -1? (c) Find the domain of f(x). (d) Find the range of f(x). (e) Is this function one-to-one? (f) Write the equation of f(x).

Analyze the function f(x) based on its graph to find key details such as f(-3), when f(x) = -1, the domain, range, and whether the function is one-to-one. Also, deduce the equation of the function from its graphical behavior.

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