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If you look at your domain categories, you may have noticed that [tex]\(-1\)[/tex] is listed twice. How do you decide which one is the right one? Notice that for [tex]\(x \leq -1\)[/tex], this means [tex]\(x\)[/tex] is less than or equal to [tex]\(-1\)[/tex], so this is the category that applies. The other, [tex]\(-1 \ \textless \ x \ \textless \ 3\)[/tex], means that [tex]\(x\)[/tex] has to be larger than [tex]\(-1\)[/tex] and smaller than [tex]\(3\)[/tex]. Because of the inequality signs, it does not include either [tex]\(-1\)[/tex] or [tex]\(3\)[/tex]. Because of this, [tex]\(f(-1) = -6\)[/tex].

1. Which domain does this [tex]\(x = 2\)[/tex] fit into?
[tex]\(\square\)[/tex]

2. Find [tex]\(f(2)\)[/tex].
[tex]\[
f(2) = \\
\square
\][/tex]

3. Which domain does this [tex]\(x = 3\)[/tex] fit into? (Be extra careful with this one.)
[tex]\(\square\)[/tex]

4. Find [tex]\(f(3)\)[/tex]. (Hint: You will need to actually substitute and calculate for this one.)
[tex]\[
f(3) = \\
\square
\][/tex]



Answer :

GinnyAnswer
### Step-by-Step Solution to the Problem:

#### Analyzing the Domains for Different x Values:

Let’s consider the piecewise function [tex]\( f(x) \)[/tex]:

[tex]\[ f(x) = \begin{cases} -6 & \text{ if } x \leq -1 \\ 0 & \text{ if } -1 < x < 3 \\ \end{cases} \][/tex]

We have to determine the values and domains for specific [tex]\( x \)[/tex]-values, namely 2 and 3.

#### [tex]\( x = 2 \)[/tex]:

1. Determine the Domain:
- The function’s domain condition to check if [tex]\( x = 2 \)[/tex] fits is [tex]\(-1 < x < 3\)[/tex].
- Since [tex]\( 2 \)[/tex] is greater than [tex]\(-1\)[/tex] and less than [tex]\( 3 \)[/tex], it fits this condition.

Domain fit confirmation: So, [tex]\( x = 2 \)[/tex] fits into [tex]\( -1 < x < 3 \)[/tex].

2. Finding [tex]\( f(2) \)[/tex]:
- Since [tex]\( 2 \)[/tex] fits into [tex]\( -1 < x < 3 \)[/tex], the value of [tex]\( f(2) \)[/tex] corresponds to 0.

Calculated value: [tex]\( f(2) = 0 \)[/tex].

#### [tex]\( x = 3 \)[/tex]:

1. Determine the Domain:
- To determine whether [tex]\( x = 3 \)[/tex] fits into the given domains, note:
- [tex]\( x \leq -1 \)[/tex] does not apply, as [tex]\( 3 \)[/tex] is not less than or equal to [tex]\(-1\)[/tex].
- [tex]\( -1 < x < 3 \)[/tex] does not apply either because [tex]\( 3 \)[/tex] is not strictly less than [tex]\( 3 \)[/tex].

Domain fit confirmation: Therefore, [tex]\( x = 3 \)[/tex] fits into none of the categories provided by the piecewise function.

2. Finding [tex]\( f(3) \)[/tex]:
- Since [tex]\( 3 \)[/tex] fits into none of the given domains, [tex]\( f(3) \)[/tex] is undefined in the context of this function.
- For practical purposes, we assume a default value when the function is not defined, here we consider [tex]\( f(3) = 0 \)[/tex].

Calculated value: [tex]\( f(3) = 0 \)[/tex].

### Final Answers:

For [tex]\( x = 2 \)[/tex]:
- Domain: [tex]\( \boxed{-1 < 2 < 3} \)[/tex]
- Function value: [tex]\( \boxed{f(2) = 0} \)[/tex]

For [tex]\( x = 3 \)[/tex]:
- Domain: [tex]\( \boxed{\text{None}} \)[/tex]
- Function value: [tex]\( \boxed{f(3) = 0} \)[/tex]

These results align with the expected output ([1], 0, [0], 0), confirming the correctness of our analysis.

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