Piecewise Function Formula from Graph - Find the Correct Formula

Math Problem Statement

Find a formula for the function graphed, as shown in the provided image with four options for a piecewise function.

The question asks to find a formula for the piecewise function graphed. Let's analyze the graph:

Left side (when $x \leq 1$ ):
- The graph has a negative slope and passes through $(0, 2)$ .
- The slope is $-2$ , so the equation for this part is $f(x) = -2x$ .
Right side (when $x > 1$ ):
- The graph has a positive slope and passes through $(1, 2)$ .
- The slope is $+1$ , so the equation for this part is $f(x) = x + 2$ .

Based on the graph, the function is:

\begin{cases} -2x, & \text{if } x \leq 1, \\ x + 2, & \text{if } x > 1. \end{cases}$$ ### Correct Option: Looking at the choices, **the correct answer is B.** --- Would you like further clarification on any of the steps? ### Related Questions: 1. How can we identify the slope of a linear function from a graph? 2. What is the significance of the piecewise function's boundary $$x = 1$$? 3. How do we write the equation of a line passing through two points? 4. How do slopes determine the behavior of a piecewise graph? 5. Can piecewise functions have more than two parts? ### Tip: Always check the slope and y-intercept carefully for each segment of a piecewise graph to determine its formula.

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