The function f is defined as f(x)=2sqrt(9-x). What is the x-coordinate of the point on the function's graph that is closest to the origin?

Question

The function  f is defined as  f(x)=2sqrt(9-x). What is the  x-coordinate of the point on the function's graph that is closest to the origin?

Bytelearn · Accepted Answer

Define Function: The function is f(x) = 2sqrt(9 - x). To find the x-coordinate of the point closest to the origin, we need to minimize the distance from the origin to the point on the graph.Distance Formula: The distance from the origin to any point (x, f(x)) on the graph is given by the distance formula: D = sqrt(x^2 + f(x)^2).Substitute and Simplify: Substitute f(x) into the distance formula: D = sqrt(x^2 + (2sqrt(9 - x))^2).Minimize Distance: Simplify the distance formula: D = sqrt(x^2 + 4(9 - x)).Expand Expression: To minimize D, we need to minimize the expression under the square root since the square root function is increasing. So we minimize x^2 + 4(9 - x).Complete the Square: Expand the expression: x^2 + 4(9 - x) = x^2 + 36 - 4x.To find the minimum, we can complete the square or take the derivative and set it to zero. Since we're looking for a simple solution, let's complete the square.Rewrite the expression as a perfect square: x^2 - 4x + 36 = (x - 2)^2 + 32.

The function $f$ is defined as $f(x)=2 \sqrt{9-x}$ . $\newline$ What is the $x$ -coordinate of the point on the function's graph that is closest to the origin?

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The function f f f is defined as f(x)=29−x f(x)=2 \sqrt{9-x} f(x)=29−x​.\newlineWhat is the x x x-coordinate of the point on the function's graph that is closest to the origin?

The function $f$ is defined as $f(x)=2 \sqrt{9-x}$ . $\newline$ What is the $x$ -coordinate of the point on the function's graph that is closest to the origin?