Question

Which of the following piecewise functions matches the graph? (1 point) Responses f(x)={2−x if x≤0, x−5 if x>0 f left parenthesis x right parenthesis equals left brace Start Rows 1st Row 2 minus x if x less-than-or-equal-to 0 2nd Row x minus 5 if x greater than 0 End Rows f(x)={−5 if x<0−x+1 if x≥0 f left parenthesis x right parenthesis equals left brace Start Rows 1st Row negative 5 if x less than 0 2nd Row negative x plus 1 if x greater-than-or-equal-to 0 End Rows f(x)={−x−1 if x<1−x+2 if x≥1 f left parenthesis x right parenthesis equals left brace Start Rows 1st Row negative x minus 1 if x less than 1 2nd Row negative x plus 2 if x greater-than-or-equal-to 1 End Rows f(x)={x+5 if x≤−22x+3 if x>−2

Answer 1

To determine which piecewise function matches the graph, we need to carefully analyze each piece of the functions provided.

1. Function 1: $$f(x)=\{\begin{array}{lll}2-x& \text{if }x\le 0x-5& \text{if }x>0\end{array}$$

   • For $x\le 0$, the function is a line with a positive slope that intersects the y-axis at $(0,2)$.
   • For $x>0$, the function is linear with a slope of 1, starting from the point $(0,-5)$ (which does not match the previous point).

2. Function 2: $$f(x)=\{\begin{array}{lll}-5& \text{if }x<0-x+1& \text{if }x\ge 0\end{array}$$

   • For $x<0$, the function is constant at -5.
   • For $x\ge 0$, the function is linear and decreases, starting from $(0,1)$.

3. Function 3: $$f(x)=\{\begin{array}{lll}-x-1& \text{if }x<1-x+2& \text{if }x\ge 1\end{array}$$

   • For $x<1$, the function has a negative slope starting from the point $(1,-2)$.
   • For $x\ge 1$, the line is still decreasing, starting from the point $(1,1)$.

4. Function 4: $$f(x)=\{\begin{array}{lll}x+5& \text{if }x\le -22x+3& \text{if }x>-2\end{array}$$

   • For $x\le -2$, the function has a positive slope starting at -2, giving a point at $(-2,3)$.
   • For $x>-2$, the function continues with a slope of 2 starting from that point.

To find the matching function, it is essential to analyze critical points, slopes, and endpoint values of the graph you are provided. Since I do not have access to the graph in question, I cannot definitively identify the matching function. However, carefully considering the slopes and points of intersection as described should allow you to determine which function corresponds to the graph. Please relate these details to the graph for clarity.