Question

1. discuss mathematical thinking describe how the graph of each piece - wise function changes when < is replaced with ≤ and ≥ is replaced with >. do the domain and range change? explain.

a. $f(x)=\begin{cases}x + 2, &\text{if }x < 2\\-x - 1, &\text{if }xgeq2end{cases}$
b. $f(x)=\begin{cases}\frac{1}{2}x+\frac{3}{2}, &\text{if }x < 1\\-x + 2, &\text{if }xgeq1end{cases}$

Explanation:

Step1: Analyze piece - wise function a

For \(f(x)=

$$\begin{cases}x + 2, &\text{if }x<2\\-x - 1, &\text{if }x\geq2\end{cases}$$

\), when \(<\) is replaced with \(\leq\) and \(\geq\) is replaced with \(>\), the function becomes \(f(x)=

$$\begin{cases}x + 2, &\text{if }x\leq2\\-x - 1, &\text{if }x>2\end{cases}$$

\). The graph of \(y = x+2\) for \(x < 2\) now includes the point \((2,4)\) and the graph of \(y=-x - 1\) for \(x\geq2\) now excludes the point \((2,-3)\). The domain remains all real numbers \((-\infty,\infty)\) because the set of all possible \(x\) - values is unchanged. The range also remains the same. For \(y=x + 2,x<2\), \(y<4\) and for \(y=-x - 1,x\geq2\), \(y\leq - 3\). After the change, for \(y=x + 2,x\leq2\), \(y\leq4\) and for \(y=-x - 1,x>2\), \(y<-3\), but the overall set of \(y\) - values is the same.

Step2: Analyze piece - wise function b

For \(f(x)=

$$\begin{cases}\frac{1}{2}x+\frac{3}{2},&\text{if }x < 1\\-x + 2,&\text{if }x\geq1\end{cases}$$

\), when \(<\) is replaced with \(\leq\) and \(\geq\) is replaced with \(>\), the function becomes \(f(x)=

$$\begin{cases}\frac{1}{2}x+\frac{3}{2},&\text{if }x\leq1\\-x + 2,&\text{if }x>1\end{cases}$$

\). The graph of \(y=\frac{1}{2}x+\frac{3}{2}\) for \(x < 1\) now includes the point \((1,2)\) and the graph of \(y=-x + 2\) for \(x\geq1\) now excludes the point \((1,1)\). The domain is still all real numbers \((-\infty,\infty)\). For the range, for \(y=\frac{1}{2}x+\frac{3}{2},x<1\), \(y<2\) and for \(y=-x + 2,x\geq1\), \(y\leq1\). After the change, for \(y=\frac{1}{2}x+\frac{3}{2},x\leq1\), \(y\leq2\) and for \(y=-x + 2,x>1\), \(y<1\), but the overall set of \(y\) - values is the same.

Answer:

For both piece - wise functions, the domain and range do not change. The graph changes in that the "open - circle" and "closed - circle" points at the break - points of the piece - wise functions are swapped.