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Chapter 1: Problem 58

Graph the function and determine the interval(s) for which \(f(x) \geq 0\). $$f(x)=x^{2}-4 x$$

Short Answer

Expert verified
The intervals for which \(f(x) \geq 0\) are \(-\infty < x \leq 0\) and \(4 \leq x < \infty\)

Step by step solution

01

Identify the Root(s) of the Function

Set the function equal to zero and solve for \(x\). \(x^{2}-4x=0\).This equation can be factored to \(x(x - 4) = 0\). Solving for \(x\), we obtain the roots: \(x = 0, 4\).
02

Graph the Function

Graph the function \(f(x)=x^{2}-4x\) using the roots identified above. The graph should be a parabola opening upwards, with the roots marking where the graph touches or crosses the x-axis.
03

Identify Intervals Where the Function is Greater Than or Equal to Zero

By visual observation from the graph, it is clear that \(f(x)\) crosses the x-axis at \(x=0\) and \(x=4\), and underneath the x-axis for \(x\) values between 0 and 4. Therefore, the function \(f(x)\) is greater than or equal to zero at the intervals \(-\infty < x \leq 0\) and \(4 \leq x < \infty\)

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