Answered

If [tex]x = -3[/tex], which domain category does it fit into?

[tex]x \leq -1[/tex]

Find [tex]f(-3)[/tex]:

Since [tex]-3 \leq -1[/tex], the answer to [tex]f(-3)[/tex] will be -6, as this is the part of the function that matches with [tex]x \leq -1[/tex]. When you have just a constant for a part of a piecewise function, any input (x-value) that falls within that domain will always yield the constant value. Thus, [tex]f(-3) = -6[/tex] and similarly, [tex]f(-4) = -6[/tex].

If [tex]x = -1[/tex], which domain category does it fit into?

Find [tex]f(-1)[/tex]:

Notice that for [tex]x \leq -1[/tex], the value -1 is included. The other category, [tex]-1 \ \textless \ x \ \textless \ 3[/tex], does not include -1 or 3. Because of the inequality signs, [tex]f(-1) = -6[/tex] as well.

If [tex]x = 2[/tex]:

Which domain category does it fit into?

Find [tex]f(2)[/tex].



Answer :

GinnyAnswer
Sure, let's break down how to find [tex]\( f(2) \)[/tex] step-by-step using the piecewise function:

1. Identify the domain:
- We have several domain categories:
- [tex]\( x \leq -1 \)[/tex]
- [tex]\( -1 < x < 3 \)[/tex]
- [tex]\( x \geq 3 \)[/tex]

2. Determine which domain [tex]\( 2 \)[/tex] fits into:
- We see that [tex]\( 2 \)[/tex] satisfies the condition [tex]\( -1 < 2 < 3 \)[/tex].
- Therefore, [tex]\( 2 \)[/tex] falls into the category [tex]\( -1 < x < 3 \)[/tex].

3. Apply the function corresponding to the domain [tex]\( -1 < x < 3 \)[/tex]:
- The function for this domain is [tex]\( f(x) = 2x + 4 \)[/tex].

4. Calculate [tex]\( f(2) \)[/tex] using the function [tex]\( f(x) = 2x + 4 \)[/tex]:
- Substituting [tex]\( x = 2 \)[/tex] into the function:
[tex]\[ f(2) = 2(2) + 4 \][/tex]

5. Perform the calculation:
- First, we do the multiplication:
[tex]\[ 2(2) = 4 \][/tex]
- Next, we add the 4:
[tex]\[ 4 + 4 = 8 \][/tex]

Therefore, [tex]\( f(2) = 8 \)[/tex].