Let's analyze the given piecewise function step by step to address the questions at hand.
### Step 1: Determine the Domain
1. Look at the function definition:
[tex]\[
y=\left\{\begin{array}{ll}
x-1 & \text { if } x<-5 \\
x-5 & \text { if }-3
2. The function [tex]\( y \)[/tex] is defined for [tex]\( x < -5 \)[/tex], [tex]\(-3 < x \leq 4\)[/tex], and [tex]\( x > 5 \)[/tex].
3. Since there are no other restrictions provided on [tex]\( x \)[/tex], the domain is:
[tex]\[
\text{All real numbers}
\][/tex]
### Step 2: Evaluate [tex]\( f(a) \)[/tex]
We will evaluate the function [tex]\( f(a) \)[/tex] for each given value of [tex]\( a \)[/tex].
#### a. [tex]\( f(8) \)[/tex]
1. Identify the interval for [tex]\( x = 8 \)[/tex]:
[tex]\[
5 < 8
\][/tex]
2. Use the corresponding piece of the function:
[tex]\[
f(8) = 8 - 8 = 0
\][/tex]
3. Therefore,
[tex]\[
f(8) = 0
\][/tex]
#### b. [tex]\( f(4) \)[/tex]
1. Identify the interval for [tex]\( x = 4 \)[/tex]:
[tex]\[
-3 < 4 \leq 4
\][/tex]
2. Use the corresponding piece of the function:
[tex]\[
f(4) = 4 - 5 = -1
\][/tex]
3. Therefore,
[tex]\[
f(4) = -1
\][/tex]
#### c. [tex]\( f(-7) \)[/tex]
1. Identify the interval for [tex]\( x = -7 \)[/tex]:
[tex]\[
-7 < -5
\][/tex]
2. Use the corresponding piece of the function:
[tex]\[
f(-7) = -7 - 1 = -8
\][/tex]
3. Therefore,
[tex]\[
f(-7) = -8
\][/tex]
#### d. [tex]\( f(5) \)[/tex]
1. Identify the interval for [tex]\( x = 5 \)[/tex]:
[tex]\[
x = 5
\][/tex]
2. Notice that [tex]\( x = 5 \)[/tex] does not fall within any interval defined in the function:
- It is not less than -5.
- It is not greater than 5.
- It is not strictly between -3 and 4.
3. Since there is no corresponding piece for [tex]\( x = 5 \)[/tex], the function value:
[tex]\[
f(5) = \text{None}
\][/tex]
### Summary of the Evaluation
1. The domain of [tex]\( f(x) \)[/tex] is:
[tex]\[
\text{all real numbers}
\][/tex]
2. The evaluations are:
- [tex]\( f(8) = 0 \)[/tex]
- [tex]\( f(4) = -1 \)[/tex]
- [tex]\( f(-7) = -8 \)[/tex]
- [tex]\( f(5) = \text{None} \)[/tex].
These results summarize our analysis and evaluations of the given piecewise function.
