Question
On a coordinate plane, a curved line crosses the x-axis at (negative 1.5, 0), the y-axis at (0, negative 2), and the x-axis at (1.5, 0).
Determine the domain and range of the given function.
The domain is
.
The range is
Determine the domain and range of the given function.
The domain is
.
The range is
Answers
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Answered by
GPT-4o
To determine the domain and range of the given function, we analyze the intersection points and the behavior implied by the description.
Given the points where the curve intersects the axes:
- The curve crosses the x-axis at \((-1.5, 0)\) and \((1.5, 0)\).
- The curve crosses the y-axis at \((0, -2)\).
### Domain:
The domain of a function is the set of all possible x-values for which the function is defined.
Since the curve crosses the x-axis at \((-1.5, 0)\) and \((1.5, 0)\), and we assume no additional constraints or vertical asymptotes given by the problem, we can infer that the function is defined at least from \(x = -1.5\) to \(x = 1.5\), inclusive.
Therefore, the domain is \([-1.5, 1.5]\).
### Range:
The range of a function is the set of all possible y-values the function can take.
Given that the curve crosses the y-axis at \((0, -2)\) and the x-axis at \((±1.5, 0)\), it is implied that \(-2\) is likely a minimum value the function attains (since \((0, -2)\) is the only given y-intercept and it is consistent with the shape described curving back to the x-axis). The function increases from \(-2\) and intersects the x-axis (where \(y = 0\)) at \((0, -2)\) to zero at \((±1.5, 0)\).
Thus, the lowest y-value the function takes is \(-2\) and the highest value is \(0\).
Therefore, the range is \([-2, 0]\).
In summary:
- The domain is \([-1.5, 1.5]\).
- The range is \([-2, 0]\).
Given the points where the curve intersects the axes:
- The curve crosses the x-axis at \((-1.5, 0)\) and \((1.5, 0)\).
- The curve crosses the y-axis at \((0, -2)\).
### Domain:
The domain of a function is the set of all possible x-values for which the function is defined.
Since the curve crosses the x-axis at \((-1.5, 0)\) and \((1.5, 0)\), and we assume no additional constraints or vertical asymptotes given by the problem, we can infer that the function is defined at least from \(x = -1.5\) to \(x = 1.5\), inclusive.
Therefore, the domain is \([-1.5, 1.5]\).
### Range:
The range of a function is the set of all possible y-values the function can take.
Given that the curve crosses the y-axis at \((0, -2)\) and the x-axis at \((±1.5, 0)\), it is implied that \(-2\) is likely a minimum value the function attains (since \((0, -2)\) is the only given y-intercept and it is consistent with the shape described curving back to the x-axis). The function increases from \(-2\) and intersects the x-axis (where \(y = 0\)) at \((0, -2)\) to zero at \((±1.5, 0)\).
Thus, the lowest y-value the function takes is \(-2\) and the highest value is \(0\).
Therefore, the range is \([-2, 0]\).
In summary:
- The domain is \([-1.5, 1.5]\).
- The range is \([-2, 0]\).
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