https://file.aiquickdraw.com/mathbot/file/1725948421880uqa1evnm.jpg

Let's analyze the given graph and answer each question in detail based on the function \( f(x) \).

### (a) The domain of \( f(x) \)

The domain of a function represents all the possible values of \( x \) for which \( f(x) \) is defined. From the graph:

- The function exists from \( x = -6 \) to \( x = 1 \) with some open circles indicating points where the function is not defined at specific \( x \)-values.
- The function is undefined at \( x = -3 \) and \( x = 0 \) (indicated by open circles at these points).

Thus, the domain of \( f(x) \) is:

\[
\text{Domain of } f(x) = (-6, -3) \cup (-3, 0) \cup (0, 1]
\]

### (b) The range of \( f(x) \)

The range is the set of all possible \( y \)-values the function takes. Based on the graph:

- The minimum \( y \)-value is \( -3 \) (reached at \( x = -1 \)).
- The maximum \( y \)-value seems to be unbounded as the graph increases for \( x > 1 \).
- The function skips some values, like it doesn’t exist at \( y = -1 \) (corresponding to \( x = 0 \)).

Thus, the range of \( f(x) \) is:

\[
\text{Range of } f(x) = [-3, 2) \cup (2, \infty)
\]

### (c) \( f(1) \)

From the graph, at \( x = 1 \), the function has a solid point at \( y = 4 \).

\[
f(1) = 4
\]

### (d) \( f(-3) \)

At \( x = -3 \), there is an open circle at \( y = 0 \), indicating that the function is not defined at \( x = -3 \).

\[
f(-3) \text{ is undefined}
\]

### (e) \( (f \circ f)(-1) \)

This represents \( f(f(-1)) \). First, we need to find \( f(-1) \):

- From the graph, \( f(-1) = -2 \).

Next, we evaluate \( f(-2) \):

- From the graph, \( f(-2) = 0 \).

Thus:

\[
(f \circ f)(-1) = f(f(-1)) = f(-2) = 0
\]

### (f) Find a nonzero value of \( x \) for which \( g(x) = \frac{1}{1 + f(x)} \) is not defined.

The expression \( g(x) \) will be undefined when the denominator is zero, i.e., when:

\[
1 + f(x) = 0 \quad \Rightarrow \quad f(x) = -1
\]

From the graph, \( f(x) = -1 \) at \( x = 0 \).

Thus, the nonzero value of \( x \) is:

\[
x = 0 \quad \text{(but since we need a nonzero value, there are none that fit this condition)}
\]

However, strictly speaking, \( x = 0 \) is the solution if non-zero wasn't specified.

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Let me know if you want further details or have any questions! Here are 5 related questions for deeper understanding:

1. How would the graph change if \( f(x) \) had a discontinuity at more points?
2. What would the domain be if \( f(x) \) was defined at \( x = 0 \)?
3. How do we calculate \( (f \circ f)(x) \) in general for composite functions?
4. How does the range of a function influence the solutions of \( g(x) \)?
5. How can we interpret the undefined points in terms of limits?

**Tip:** Always check for open and closed points in graphs to identify where a function is defined or undefined.

Given the graph of a function f(x) below, answer the following questions: (a) The domain of f(x) is (b) The range of f(x) is (c) f(1) = (d) f(-3) = (e) (f ∘ f)(-1) = (f) Find a nonzero value of x for which g(x) = 1/(1 + f(x)) is not defined.

Learn to determine the domain and range of functions, evaluate composite functions, and identify undefined expressions using a graph. This problem involves key mathematical concepts such as functions, domain and range, and composite functions, suitable for students in Grades 9-12.

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