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Chapter 2: Problem 138

Define a piecewise function on the intervals \((-\infty, 2],(2,5)\) and \([5, \infty)\) that does not "jump" at 2 or 5 such that one piece is a constant function, another piece is an increasing function, and the third piece is a decreasing function.

Short Answer

Expert verified
A suitable piecewise function could be: \[ f(x) = \begin{cases} 1 & \text{for } x \leq 2 \\ x - 1 & \text{for } 2 < x < 5 \\ 9 - x & \text{for } x \geq 5 \end{cases}\]

Step by step solution

01

Define first piece

Choose a constant function for the interval \((-\infty, 2]\). The simplest constant function is a horizontal line. Let's make it \(f(x) = 1\). This function is constant for all values of x, so it's definitely constant in the interval \((-\infty, 2]\).
02

Define second piece

In the interval \((2,5)\), the function should be increasing. Here, the simplest choice is a linear function, whose general form is \(f(x) = mx + b\). We already know that \(f(2) = 1\), so we choose a function that passes through the point (2,1). Let's make it \(f(x) = x - 1\), which is increasing and \(f(2) = 1\)
03

Define third piece

For the interval \([5,\infty)\) the function should be decreasing. Again, the simplest choice is a linear function. We know that the function should pass through the point (5,4) (from the second piece we defined), so let's choose a function with a negative slope, like \(f(x) = 9 - x\)

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Graph \(y_{1}=x^{2}-2 x, y_{2}=x,\) and \(y_{3}=y_{1} \div y_{2}\) in the same \([-10,10,1]\) by \([-10,10,1]\) viewing rectangle. Then use the TRACE \(]\) feature to trace along \(y_{3} .\) What happens at \(x=0 ?\) Explain why this occurs.

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