To compare the intercepts and intersection points of the functions \( f(x) = 2\sqrt{x} \) and \( g(x) = 2\sqrt[3]{x} \), we can analyze each function.
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Find the intercepts:
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For \( f(x) = 2\sqrt{x} \):
- x-intercept: Set \( f(x) = 0 \): \[ 2\sqrt{x} = 0 \implies \sqrt{x} = 0 \implies x = 0. \] Thus, the x-intercept is \( (0, 0) \).
- y-intercept: \( f(0) = 2\sqrt{0} = 0 \). Thus, the y-intercept is also \( (0, 0) \).
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For \( g(x) = 2\sqrt[3]{x} \):
- x-intercept: Set \( g(x) = 0 \): \[ 2\sqrt[3]{x} = 0 \implies \sqrt[3]{x} = 0 \implies x = 0. \] Thus, the x-intercept is \( (0, 0) \).
- y-intercept: \( g(0) = 2\sqrt[3]{0} = 0 \). Thus, the y-intercept is also \( (0, 0) \).
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Find intersection points:
We need to set \( f(x) \) equal to \( g(x) \): \[ 2\sqrt{x} = 2\sqrt[3]{x}. \] Dividing both sides by 2 (as long as \( x \geq 0 \)), we get: \[ \sqrt{x} = \sqrt[3]{x}. \] Raising both sides to the power of 6 (to eliminate the roots): \[ x^3 = x^2. \] Rearranging gives: \[ x^3 - x^2 = 0 \implies x^2(x - 1) = 0. \] Thus, \( x = 0 \) or \( x = 1 \).
- At \( x = 0 \): \( f(0) = g(0) = 0 \), so the point is \( (0, 0) \).
- At \( x = 1 \): \( f(1) = 2\sqrt{1} = 2 \) and \( g(1) = 2\sqrt[3]{1} = 2 \), so the point is \( (1, 2) \).
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Conclusion:
Both functions share their x- and y-intercepts at \( (0, 0) \) and they also intersect at the point \( (1, 2) \).
Thus, the correct response is: Both graphs share their x-and y-intercepts at (0, 0) and they also intersect at the point (1, 2).