Let's solve the given problem step-by-step, following the provided conditions and clues.
### Step 1: Interpret the Information
The problem statement provides some specific conditions and values:
- For [tex]\( x = 2 \)[/tex], we have [tex]\( f(2) = 3 \)[/tex].
Given the intervals or domains for the function, we need to determine the appropriate domain for [tex]\( x = 3 \)[/tex]:
- The first domain provided is [tex]\( -1 < x < 3 \)[/tex].
- The second domain is [tex]\( x \geq 3 \)[/tex].
### Step 2: Determine the Domain for [tex]\( x = 3 \)[/tex]
Next, we need to decide which of these domains includes [tex]\( x = 3 \)[/tex]:
- The domain [tex]\( -1 < x < 3 \)[/tex] does not include 3, because it is strictly less than 3.
- The domain [tex]\( x \geq 3 \)[/tex] does include [tex]\( x = 3 \)[/tex].
Therefore, [tex]\( x = 3 \)[/tex] fits into the domain [tex]\( x \geq 3 \)[/tex].
### Step 3: Find [tex]\( f(3) \)[/tex]
Knowing that [tex]\( x = 3 \)[/tex] fits into the domain [tex]\( x \geq 3 \)[/tex], we use the hint provided. The hint states that we will need to find [tex]\( f(3) \)[/tex] specifically. The given information tells us:
- [tex]\( f(3) = 3 \)[/tex].
So, at [tex]\( x = 3 \)[/tex], the function value is:
- [tex]\( f(3) = 3 \)[/tex].
### Final Results
- For [tex]\( x = 2 \)[/tex]: [tex]\( f(2) = 3 \)[/tex].
- For [tex]\( x = 3 \)[/tex]: [tex]\( f(3) = 3 \)[/tex].
Hence, the solution to the problem is:
[tex]\[ f(3) = 3 \][/tex]
