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Chapter 1: Problem 70

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 \((- \infty,0] \cup [4, \infty)\).

Step by step solution

01

Identify the roots of the equation

Start by finding the roots of the equation by setting \(f(x)\) equal to 0, i.e. solving the equation \(x^2-4x=0\). This can be done by factoring out the greatest common factor, which in this case is \(x\). We get \(x(x-4)=0\). The solutions to this equation (the x-values that make the equation true) are \(x=0\) and \(x=4\). These will be the x-intercepts of the graph.
02

Construct the graph

Now we can graph the function using the roots and the vertex. The function is a parabola that opens upward (since the coefficient of \(x^2\) is positive 1). The roots we found in Step 1 tell us it crosses the x-axis at (0,0) and (4,0). Moreover, because we know the graph opens upwards and fails to cross the x-axis in between roots, the vertex will be in between the roots at x=2. If we plug this into our equation, we find that the y-coordinate (and thus the minimum value of the function) is \(-(4^2)/4=-4\). So we can plot the point (2, -4) as the vertex.
03

Identify the intervals where \(f(x) \geq 0\)

Next, we need to find the intervals for which \(f(x) \geq 0\). From the graph, it's clear that this is true from x=-infinity to x=0 and from x=4 to x=infinity. The function is exactly 0 at x=0 and x=4, but greater than 0 for all other values in these intervals. So the intervals where \(f(x) \geq 0\) are \((- \infty,0] \cup [4, \infty)\) in interval notation.

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