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To solve this problem, we need to identify the subfunctions \( f(x) \) for the piecewise function and their corresponding subdomains by analyzing the given graph.

### Analyzing the Graph:

1. **First Segment (a):**
   - The line segment starts at \( x = -10 \) and ends at \( x = 0 \).
   - The line appears to have a negative slope and crosses the y-axis at \( y = 0 \).
   - To find the equation of this line, we use two points from the graph: \( (-10, 5) \) and \( (0, -5) \).

   The slope \( m \) of the line is:
   \[
   m = \frac{-5 - 5}{0 - (-10)} = \frac{-10}{10} = -1
   \]
   The equation of the line is then:
   \[
   y = -x - 5
   \]
   So, \( f(x) = -x - 5 \) for \( x \) in the subdomain \( -10 \leq x \leq 0 \).

2. **Second Segment (b):**
   - The line segment starts at \( x = 0 \) and ends at \( x = 10 \).
   - This line has a positive slope.
   - Using points \( (0, -5) \) and \( (10, 10) \), we find the slope \( m \):

   The slope \( m \) is:
   \[
   m = \frac{10 - (-5)}{10 - 0} = \frac{15}{10} = 1.5
   \]
   The equation of the line is:
   \[
   y = 1.5x - 5
   \]
   So, \( f(x) = 1.5x - 5 \) for \( x \) in the subdomain \( 0 \leq x \leq 10 \).

### Answer:

- For the subfunction \( a \):
  \[
  f(x) = -x - 5, \quad -10 \leq x \leq 0
  \]

- For the subfunction \( b \):
  \[
  f(x) = 1.5x - 5, \quad 0 \leq x \leq 10
  \]

Would you like further details or have any questions about this solution?

### Related Questions:
1. How do you determine the slope of a line between two points?
2. What is the general form of a linear equation?
3. How can you verify if a given point lies on a particular line?
4. What are the steps to write the equation of a line given two points?
5. What is a piecewise function, and how is it different from a regular function?

### Tip:
When determining the equation of a line from a graph, always ensure you correctly identify two clear points on the line to calculate the slope accurately.

Suppose the piecewise function f(x) = a if a is shown below. Enter the subfunction (f(x) =) and subdomain for a and b in order and separated by a comma.

Learn how to solve a piecewise function with two linear segments by identifying the subfunctions and subdomains. This problem involves key concepts such as slope calculation and linear equations, suitable for high school students in grades 10-12.

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