Question
1.Find the domain and range of the inverse of arelation R=(x,y): y < -x2 and y 2 -1)
Answers
GPT-4o mini
To find the domain and range of the inverse of the relation \( R = \{ (x, y) : y < -x^2 \text{ and } y < -1 \} \), we first need to understand the original relation.
The relation \( R \) describes a region in the Cartesian plane where \( y \) is less than both \( -x^2 \) (which is a downward-opening parabola) and \( -1 \) (which is a horizontal line).
### Step 1: Analyzing the original relation \( R \)
1. **The parabola**: The inequality \( y < -x^2 \) defines the region below the parabola that opens downwards.
2. **The horizontal line**: The inequality \( y < -1 \) defines the region below the line \( y = -1 \).
The combined inequalities \( y < -x^2 \) and \( y < -1 \) means we are looking at the region below both the parabola and the horizontal line.
### Step 2: Visualizing the relation
- The vertex of the parabola \( y = -x^2 \) is located at the origin (0, 0) and the parabola opens downwards.
- The line \( y = -1 \) intersects the \( y \)-axis at (0, -1).
### Step 3: Finding domain and range of \( R \)
**Domain of \( R \)**:
The domain of the relation \( R \) is all real numbers, as \( x \) can take any value.
Thus, the domain of \( R \) is:
\[
\text{Domain}(R) = (-\infty, \infty)
\]
**Range of \( R \)**:
For the range, since the region is defined by \( y < -1 \) and \( y < -x^2 \), it implies \( y \) can take any value less than -1.
Thus, the range of \( R \) is:
\[
\text{Range}(R) = (-\infty, -1)
\]
### Step 4: Finding the domain and range of the inverse relation \( R^{-1} \)
The inverse relation \( R^{-1} \) consists of pairs where the \( x \)- and \( y \)-coordinates are swapped.
1. **Domain of \( R^{-1} \)**: This corresponds to the range of \( R \).
\[
\text{Domain}(R^{-1}) = \text{Range}(R) = (-\infty, -1)
\]
2. **Range of \( R^{-1} \)**: This corresponds to the domain of \( R \).
\[
\text{Range}(R^{-1}) = \text{Domain}(R) = (-\infty, \infty)
\]
### Conclusion
- The domain of the inverse relation \( R^{-1} \) is \( (-\infty, -1) \).
- The range of the inverse relation \( R^{-1} \) is \( (-\infty, \infty) \).
The relation \( R \) describes a region in the Cartesian plane where \( y \) is less than both \( -x^2 \) (which is a downward-opening parabola) and \( -1 \) (which is a horizontal line).
### Step 1: Analyzing the original relation \( R \)
1. **The parabola**: The inequality \( y < -x^2 \) defines the region below the parabola that opens downwards.
2. **The horizontal line**: The inequality \( y < -1 \) defines the region below the line \( y = -1 \).
The combined inequalities \( y < -x^2 \) and \( y < -1 \) means we are looking at the region below both the parabola and the horizontal line.
### Step 2: Visualizing the relation
- The vertex of the parabola \( y = -x^2 \) is located at the origin (0, 0) and the parabola opens downwards.
- The line \( y = -1 \) intersects the \( y \)-axis at (0, -1).
### Step 3: Finding domain and range of \( R \)
**Domain of \( R \)**:
The domain of the relation \( R \) is all real numbers, as \( x \) can take any value.
Thus, the domain of \( R \) is:
\[
\text{Domain}(R) = (-\infty, \infty)
\]
**Range of \( R \)**:
For the range, since the region is defined by \( y < -1 \) and \( y < -x^2 \), it implies \( y \) can take any value less than -1.
Thus, the range of \( R \) is:
\[
\text{Range}(R) = (-\infty, -1)
\]
### Step 4: Finding the domain and range of the inverse relation \( R^{-1} \)
The inverse relation \( R^{-1} \) consists of pairs where the \( x \)- and \( y \)-coordinates are swapped.
1. **Domain of \( R^{-1} \)**: This corresponds to the range of \( R \).
\[
\text{Domain}(R^{-1}) = \text{Range}(R) = (-\infty, -1)
\]
2. **Range of \( R^{-1} \)**: This corresponds to the domain of \( R \).
\[
\text{Range}(R^{-1}) = \text{Domain}(R) = (-\infty, \infty)
\]
### Conclusion
- The domain of the inverse relation \( R^{-1} \) is \( (-\infty, -1) \).
- The range of the inverse relation \( R^{-1} \) is \( (-\infty, \infty) \).