Question

Children’s cough syrup has dosage guidelines based on a child’s age. In the following piecewise function, x represents the child’s age in years and f(x) represents the dosage in fluid ounces. f(x)=⎧⎩⎨⎪⎪0.5 if 1≤x≤31.0 if 3<x≤60.5x if x>6 Does the graph accurately represent the piecewise function? Why or why not?

Answer 1

To determine whether the graph accurately represents the given piecewise function, we need to analyze the function definition and see if the graph meets its criteria.

The piecewise function is defined as follows:

1. \( f(x) = 0.5 \) if \( 1 \leq x \leq 3 \)
2. \( f(x) = 1.0 \) if \( 3 < x \leq 6 \)
3. \( f(x) = 0.5x \) if \( x > 6 \)

Now let's break down the function:

1. **Interval \( 1 \leq x \leq 3 \)**:
- For ages between 1 and 3 (inclusive), the dosage is constant at 0.5 fluid ounces. The graph should display a horizontal line at \( y = 0.5 \) from \( x = 1 \) to \( x = 3 \).

2. **Interval \( 3 < x \leq 6 \)**:
- For ages greater than 3 up to 6, the dosage is a constant 1.0 fluid ounces. The graph should show a horizontal line at \( y = 1.0 \) from \( x = 3 \) (not including 3) to \( x = 6 \) (inclusive).

3. **Interval \( x > 6 \)**:
- For ages greater than 6, the dosage follows the linear function \( f(x) = 0.5x \). The graph should reflect the line starting from \( x = 6 \) (where it coincides with the value 3.0) and continue upwards based on \( f(x) = 0.5x \).

### Checking Graph Representation:
- **At \( x = 1 \) to \( x = 3 \)**: The graph should be a horizontal line at \( y = 0.5 \).
- **At \( x = 3 \)**: The graph should show an open circle at \( (3, 0.5) \) since 3 is not included, and then be a horizontal line at \( y = 1.0 \) from \( (3, 1.0) \) to \( (6, 1.0) \) (including 6).
- **At \( x > 6 \)**: The graph should start from the point \( (6, 3) \) and then proceed as a line with a slope of 0.5.

### Conclusion:
If the graph displays these characteristics accurately, then it correctly represents the piecewise function. If there are discrepancies, such as incorrect intervals, discontinuities where there should be open circles, or an incorrect portrayal of the linear function for ages over 6, then the graph would not accurately represent the piecewise function.

Without the actual visual graph to reference, I cannot definitively say if the graph is accurate or not, but you can now analyze it against these criteria to check for its correctness.