MATH 1513: Increasing and Decreasing Functions Study Notes

Section 2.2 Overview and Homework Context

• Course Information: MATH 1513, Section 2.2 (FA26)
• Topic Focus: Increasing, Decreasing, and Constant Functions; Difference Quotient
• Instructor / Portal Reference: Dittenber
• Exercise Reference: Question 15 (Section 2.2.99, Part 1 of 2)
• Assignment Metrics:
  • Homework Score: $68.75\%$ ($13.75$ out of $20$ points)
  • Question Value: $0$ out of $1$ point
• Timestamp Data: 8/30/2026 at 4:44 PM

Mathematical Definitions of Monotonicity

• Increasing Functions:

  • A function $f$ is increasing on an open interval $I$ if, for all $x_1, x_2 \in I$:     $\text{If } x_1 < x_2 \text{, then } f(x_1) < f(x_2)$
  • On a graph, an increasing function slopes upward from left to right as the independent variable increases.

• Decreasing Functions:

  • A function $f$ is decreasing on an open interval $I$ if, for all $x_1, x_2 \in I$:     $\text{If } x_1 < x_2 \text{, then } f(x_1) > f(x_2)$
  • On a graph, a decreasing function slopes downward from left to right as the independent variable increases.

• Constant Functions:

  • A function $f$ is constant on an open interval $I$ if, for all $x_1, x_2 \in I$:     $f(x_1) = f(x_2)$
  • On a graph, a constant function forms a horizontal line segment with a slope of zero.

• Interval Representation:

  • Intervals describing behavior (increasing, decreasing, or constant) are specified using open interval notation $(a, b)$, representing the domain $x$-values across which the trend holds, excluding the boundary endpoints.

Percent Body Fat vs. Age Analysis

• Biological Context:

  • Human aging is accompanied by predictable body composition changes, notably an increase in overall body fat percentage and a reduction in muscle mass.

• Graph Structure:

  • Title: Percent Body Fat in Adults
  • Independent Variable ($x$-axis): Age (measured in years), plotted from $x = 25$ to $x = 75$, with major tick marks located at $x = 25$, $x = 50$, and $x = 75$
  • Dependent Variable ($y$-axis): Percent Body Fat (measured as a percentage), with major tick marks labeled at $y = 20$, $y = 30$, and $y = 40$
  • Data Series: The graph displays two distinct curves labeled Women and Men

• Behavior of Women's Body Fat Curve:

  • Starting at age $x = 25$, the curve representing percent body fat in adult women exhibits a continuous positive slope as age increases up to $x = 75$
  • At no point between age $25$ and age $75$ does the curve slope downward or flatten out completely

Step-by-Step Question Solution and Interval Evaluation

• Problem Statement: State the interval on which the graph giving the percent body fat in women is increasing.

• Evaluation of Answer Choices:

  • Option A: $(25, 75)$
  • Evaluation: Correct. The curve representing percent body fat in adult women rises consistently across the entire domain of age $25$ to age $75$. Thus, the graph is increasing on the open interval $(25, 75)$.
  • Option B: $(25, 35)$
  • Evaluation: Incorrect. This interval represents only a sub-portion of the overall increasing trend, which continues past age $35$ all the way up to age $75$
  • Option C: $(35, 75)$
  • Evaluation: Incorrect. This interval omits the initial increasing portion occurring between age $25$ and age $35$
  • Option D: $(20, 26)$
  • Evaluation: Incorrect. These numbers correspond to approximate $y$--axis values or misread axis ranges rather than the proper domain range on the $x$--axis

• Conclusion:

  • The graph giving the percent body fat in women is increasing on the interval $(25, 75)$