The problem is asking for the piecewise-defined function that describes the graph shown.

Analysis of the Graph

From the graph, we see two different line segments:

1. For $x \leq 2$: This part appears to be a linear function, decreasing from left to right. We can determine the equation by using the slope-intercept form $y = mx + b$.
2. For $x > 2$: This part is another linear function, increasing from left to right.

Steps to find the equations:

1. Left segment ($x \leq 2$): Identify two points, calculate the slope $m$, and use the slope-intercept form to find the equation.
2. Right segment ($x > 2$): Similarly, find the slope using two points and determine the equation.

Determining the correct answer choice

Now, let's fill in the equations for each segment in the provided options to match the graph.

• Option A: It suggests a format of $f(x)$ defined separately for $x \leq c$ and $x > c$. Verify the corresponding linear equations fit the segments on the graph.
• Option B: Similarly, verify against the segments.
• Option C: Check the provided conditions and linear forms.
• Option D: This suggests a defined range, $5 \leq x \leq c$. This is likely not correct given the graph's layout.

I will calculate the equations for each segment to match them with the options provided.The equations for each segment of the piecewise function are:

1. For $x \leq 2$: The equation is $y = 2 - 0.5x$.
2. For $x > 2$: The equation is $y = 0.5x$.

Now let's match these equations with the given options:

• Option A:

  • $f(x) =$ (left segment) if $x \leq 2$: This would be $2 - 0.5x$.
  • $f(x) =$ (right segment) if $x > 2$: This would be $0.5x$.

• Option B, C, and D do not fit the conditions for $x = 2$ properly based on their given inequalities and are likely not correct.

Therefore, Option A seems to be the correct choice as it matches the derived equations for each segment of the function.

Would you like more details on this solution, or do you have any other questions?

Related Questions

1. How do you find the slope of a line segment given two points?
2. What are piecewise functions and how are they represented?
3. How do you identify different segments in a piecewise function?
4. Can you describe other examples of piecewise functions with more than two segments?
5. How do discontinuities in piecewise functions work?

Tip: When dealing with piecewise functions, always check each segment's boundaries to ensure continuity or identify any jump discontinuities.