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A function ƒ is odd, and ƒ(x) = x2 for all x >0. Sketch the graph of this function and then write the rule of this function as a single formula.

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Answer:

The function ƒ is odd, which means that if x is a real number and ƒ(x) = 0, then x must be a real number and ƒ(−x) = −ƒ(x).

For x > 0, ƒ(x) = x2.

The graph of the function ƒ is a parabola that opens upward.

The rule of the function can be written as:

y = ƒ(x) = x2

This function is a parabola that opens upward and has a vertex at the origin. The vertex of the parabola is the point where the parabola intersects the x-axis. The y-coordinate of the vertex is given by the formula:

y-coordinate of vertex = -b/2a

where a and b are the coefficients of the x^2 term in the equation of the parabola.

In this case, a = 1 and b = 1, so the y-coordinate of the vertex is:

y-coordinate of vertex = -1/2

The x-coordinate of the vertex is not determined by the given information, but it can be calculated by equating the x-coordinate of the vertex to the y-coordinate of the vertex:

x-coordinate of vertex = -1/2

Therefore, the graph of the function ƒ is a parabola that opens upward, with a vertex at the origin and a y-coordinate of -1/2. The rule of the function is y = x^2.

User Kristian Frost
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