It seems you are referring to two different concepts: the first part discusses a circle centered at the origin with a radius of 4 units, while the second part appears to describe a mapping of integer values to specific outputs. Let's break this down.
Circle P:
The equation for a circle centered at the origin (0,0) with a radius of 4 units is given by:
\[ x^2 + y^2 = 16 \]
This is derived from the general formula for a circle \((x-h)^2 + (y-k)^2 = r^2\), where \( (h, k) \) is the center of the circle and \( r \) is the radius. In this case:
- Center \((h,k) = (0,0)\)
- Radius \(r = 4\) so \(r^2 = 16\)
The domain of the circle in the context of a vertical line test (to graph it) indicates that for every x in the range of -4 to 4, there are two corresponding y values (which confirms that the circle is not a function).
Mapping:
The second part of your query provides mappings from integers to specific outputs:
- \(0 \to 3\)
- \(1 \to 7\)
- \(2 \to 11\)
- \(3 \to 15\)
This seems to suggest a function where the input value is an integer, and the output is a corresponding value. A pattern can be observed within the outputs; it looks like the mapping could represent a linear relationship between the input (x) and output (y).
Indeed, the outputs follow the linear pattern, which can be analyzed as follows:
- Finding the pattern:
- From 0 to 3: \(3 - 0 = 3\)
- From 1 to 7: \(7 - 3 = 4\)
- From 2 to 11: \(11 - 7 = 4\)
- From 3 to 15: \(15 - 11 = 4\)
If we look closely, the output increases by 4 for the last three values. It's useful to express the output explicitly in a formula.
- Finding a linear equation: You might notice that for every incremental increase of 1 in x after x=0, we increase the output by 4. Therefore, a potential formula might look like:
\[ f(x) = 4x + 3 \]
Verifying with your points:
- For \(x = 0\): \(f(0) = 4(0) + 3 = 3\)
- For \(x = 1\): \(f(1) = 4(1) + 3 = 7\)
- For \(x = 2\): \(f(2) = 4(2) + 3 = 11\)
- For \(x = 3\): \(f(3) = 4(3) + 3 = 15\)
Conclusion:
You have a circle defined by \(x^2 + y^2 = 16\) and a mapping of integers to outputs defined by the function \(f(x) = 4x + 3\) for \(x = 0, 1, 2, 3\). Keep in mind that the circle is not a function due to the vertical line test, while the mapping you've identified does represent a valid function over the specified domain.
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