To find the domain and range of the region [tex]\( R \)[/tex] defined by the inequalities [tex]\( y \leq x + 2 \)[/tex], [tex]\( y \geq 2 - x \)[/tex], and [tex]\( y \geq 0 \)[/tex], follow these steps:
### Step 1: Analyze Boundaries and Intersection Points
Let's understand the boundaries given by the inequalities:
1. [tex]\( y \leq x + 2 \)[/tex]
2. [tex]\( y \geq 2 - x \)[/tex]
3. [tex]\( y \geq 0 \)[/tex]
#### Intersection Points:
To find the region where all these inequalities hold, we first need to identify the points where the lines intersect.
##### Intersection of [tex]\( y = x + 2 \)[/tex] and [tex]\( y = 2 - x \)[/tex]:
Set the equations equal to each other:
[tex]\[ x + 2 = 2 - x \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x + x = 2 - 2 \][/tex]
[tex]\[ 2x = 0 \][/tex]
[tex]\[ x = 0 \][/tex]
Now, find the corresponding [tex]\( y \)[/tex] value:
[tex]\[ y = 0 + 2 = 2 \][/tex]
Thus, the lines intersect at the point [tex]\((0, 2)\)[/tex].
### Step 2: Determine the Valid x and y Ranges which Satisfy all Inequalities
#### Domain (x-values):
Consider the inequality [tex]\( y \leq x + 2 \)[/tex]. For any [tex]\( y \)[/tex] that satisfies this, [tex]\( x \)[/tex] can extend from a left limit to infinity:
1. [tex]\( y \geq 2 - x \)[/tex] forces [tex]\( x \)[/tex] to be non-negative for [tex]\( y \geq 0 \)[/tex] to hold true.
2. As [tex]\( y \geq 2 - x \)[/tex], when [tex]\( y = 0 \)[/tex], [tex]\( 0 \geq 2 - x \Rightarrow x \geq 2 - y \Rightarrow x \geq 0 \)[/tex].
Therefore, the domain is:
[tex]\[ x \in [0, \infty) \][/tex]
#### Range (y-values):
Consider the inequality [tex]\( y \geq 2 - x \)[/tex]. For any [tex]\( x \)[/tex]:
1. [tex]\( y \leq x + 2 \)[/tex].
2. [tex]\( y \geq 0 \)[/tex] directly gives a lower bound on [tex]\( y \)[/tex].
Hence, the range is:
[tex]\[ y \in [0, \infty) \][/tex]
### Conclusion:
Combining these conclusions, we find that for the region [tex]\( R \)[/tex]:
- Domain: All [tex]\( x \)[/tex] values from [tex]\( 0 \)[/tex] to [tex]\( \infty \)[/tex], i.e., [tex]\( [0, \infty) \)[/tex].
- Range: All [tex]\( y \)[/tex] values from [tex]\( 0 \)[/tex] to [tex]\( \infty \)[/tex], i.e., [tex]\( [0, \infty) \)[/tex].
Thus, the domain and range are:
[tex]\[ \text{Domain: } x \in [0, \infty), \][/tex]
[tex]\[ \text{Range: } y \in [0, \infty). \][/tex]
Hence, we can summarize the result as:
[tex]\[ (x_{\text{min}}, x_{\text{max}}, y_{\text{min}}, y_{\text{max}}) = (0, \infty, 0, \infty) \][/tex]
