Problem Analysis
The given function is a piecewise function defined as:
\[
f(x) =
\begin{cases}
x + 3 & \text{if } x > 0 \\
-x & \text{if } x \leq 0
\end{cases}
\]
We are also provided with a graph that appears to represent this function. The task is to analyze the function and verify if the graph matches the given definition.
Step-by-Step Solution
#### 1.
Understanding the Piecewise Function
The function \( f(x) \) is defined in two parts:
- For \( x > 0 \): \( f(x) = x + 3 \)
- For \( x \leq 0 \): \( f(x) = -x \)
#### 2.
Analyzing Each Part of the Function
##### (a)
For \( x > 0 \):
- The function is \( f(x) = x + 3 \).
- This is a linear function with a slope of 1 and a y-intercept of 3.
- For positive values of \( x \), the graph will be a straight line starting from the point \( (0, 3) \) and extending upwards to the right.
##### (b)
For \( x \leq 0 \):
- The function is \( f(x) = -x \).
- This is also a linear function with a slope of -1 and passing through the origin \( (0, 0) \).
- For non-positive values of \( x \) (i.e., \( x \leq 0 \)), the graph will be a straight line starting from the origin and extending upwards to the left.
#### 3.
Combining the Two Parts
- At \( x = 0 \):
- From the definition, \( f(0) = -0 = 0 \). So, the point \( (0, 0) \) is included in the graph.
- For \( x > 0 \), the function starts at \( (0, 3) \) but does not include the point \( (0, 3) \) because the condition is \( x > 0 \).
#### 4.
Matching the Graph
- The graph shows:
- A line with a positive slope starting from \( (0, 3) \) for \( x > 0 \). This matches \( f(x) = x + 3 \).
- A line with a negative slope starting from \( (0, 0) \) for \( x \leq 0 \). This matches \( f(x) = -x \).
- The point \( (0, 0) \) is included (closed circle), and the point \( (0, 3) \) is not included (open circle), which is consistent with the piecewise definition.
Conclusion
The graph accurately represents the given piecewise function \( f(x) \).
\[
\boxed{\text{The graph matches the function.}}
\]
Parent Tip: Review the logic above to help your child master the concept of piecewise.
