Question

exercises 99 - 104
percent body fat in adults
source: thompson et al., the science of nutrition, benjamin cummings, 2008

1. state the intervals on which the graph giving the percent body fat in women is increasing and decreasing.
2. state the intervals on which the graph giving the percent body fat in men is increasing and decreasing.
3. for what age does the percent body fat in women reach a maximum? what is the percent body fat for that age?
4. at what age does the percent body fat in men reach a maximum? what is the percent body fat for that age?
5. use interval notation to give the domain and range for the graph of the function for women.
6. use interval notation to give the domain and range for the graph of the function for men.

Explanation:

Step1: Analyze women's graph for increasing - decreasing intervals

For women, observe the slope of the graph. The graph is increasing when the slope is positive and decreasing when the slope is negative.
The graph of percent body - fat in women is increasing on the interval $(25,45)$ and decreasing on the intervals $(0,25)$ and $(45,75)$.

Step2: Analyze men's graph for increasing - decreasing intervals

For men, observe the slope of the graph. The graph of percent body - fat in men is increasing on the interval $(25,55)$ and decreasing on the intervals $(0,25)$ and $(55,75)$.

Step3: Find women's maximum

Locate the highest point on the women's graph. The percent body - fat in women reaches a maximum at age $45$ and the percent body - fat at that age is $38\%$.

Step4: Find men's maximum

Locate the highest point on the men's graph. The percent body - fat in men reaches a maximum at age $55$ and the percent body - fat at that age is $26\%$.

Step5: Find women's domain and range

The domain is the set of all possible input values (ages). For women, the domain in interval notation is $[0,75]$. The range is the set of all possible output values (percent body - fat). The minimum value of percent body - fat for women is approximately $32\%$ and the maximum is $38\%$, so the range is $[32,38]$.

Step6: Find men's domain and range

The domain for men in interval notation is $[0,75]$. The minimum value of percent body - fat for men is approximately $22\%$ and the maximum is $26\%$, so the range is $[22,26]$.

Answer:

1. Increasing: $(25,45)$; Decreasing: $(0,25),(45,75)$
2. Increasing: $(25,55)$; Decreasing: $(0,25),(55,75)$
3. Age: $45$; Percent body - fat: $38\%$
4. Age: $55$; Percent body - fat: $26\%$
5. Domain: $[0,75]$; Range: $[32,38]$
6. Domain: $[0,75]$; Range: $[22,26]$