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Chapter 1: Problem 55

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

Short Answer

Expert verified
The interval for which the function \(f(x)\) is greater than or equal to zero is \([- \infty, 4]\)

Step by step solution

01

Understand the function

The function \(f(x) = 4 - x\) is a linear function. Linear functions have a degree one and they typically take the form \(f(x) = mx + b\) where m is the slope and b is the y-intercept. For this particular function, the slope is -1 (opposite of x鈥檚 coefficient) and the y-intercept is 4 (constant).
02

Plot the function

The function \(f(x) = 4 - x\) can be plotted on a coordinate plane. Start by plotting the y-intercept, which is at the point (0, 4). From the y-intercept, knowing that the slope is -1, this means each time x increases by 1 unit, y (or \(f(x)\)) decreases by 1 unit. So, the line can be drawn using the y-intercept and by following the slope.
03

Determine where \(f(x) \geq 0\)

The point where \(f(x)\) is equal to zero gives a point of reference for determining the required interval. For the function \(f(x) = 4 - x\), set it equal to zero to solve for x.\n\(4 - x = 0 \Longrightarrow x = 4\)\nPlot x = 4 on the graph. As the slope of the function is negative, it means that the function decreases as x increases. So, for x values to the right of x = 4 (i.e., for \(x > 4\)), \(f(x) < 0\) and for x values to the left of x = 4 (i.e., for \(x \leq 4\)), \(f(x) \geq 0\). Therefore, the required interval is \([- \infty, 4]\).

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