Monotonicity, Domain/Range, and Continuity – Notes

Reading function notation and simple mappings

• When a new notation is introduced for a function in a table or transcript, the first component shown is the input (the x-value) and the second is the output (the y-value). In other words, the first group corresponds to x and the second group to y.
• To evaluate a function from a table or mapping:
  • If you see a pair like
  • (0, 0) then the input is 0 and the output is 0, so $f(0)=0$.
  • If you see a pair like (3, 100) then $f(3)=100$.
• Example from the transcript:
  • To determine $f(0)$, look for x = 0 in the table; since the pair is 0 → 0, we have $f(0)=0$.
  • To determine $f(3)$, find the pair with x = 3; here 3 maps to 100, so $f(3)=100$.
• Graph-based evaluation:
  • For the graph shown to the right, if you are asked for values like $f(0)$ and $f(6)$, you read the y-values at those x-coordinates on the graph. If the graph is a constant function, then both $f(0)$ and $f(6)$ equal the constant value.
• Quick intuitive note on constant functions:
  • A constant function has the form $f(x)=c$ for all x in its domain, so the output does not depend on the input.

Interpreting a graph: monotonicity and open intervals

• Open intervals and domain focus:
  • The phrase “open intervals of the domain” means you look at x-values and identify where the graph is increasing, decreasing, or constant on intervals that do not include their endpoints (endpoints are not included).
  • Always focus on x-values when describing these intervals, not the y-values.
• How to read the monotonicity from left to right:
  • Decreasing: as you move right, the graph goes down.
  • Increasing: as you move right, the graph goes up.
  • Constant: the graph stays flat.
• A concrete iteration from the transcript:
  • Decreasing on the left:
  • Interval: $(-\,<br /> ablafty, 0)$
  • Endpoints are open (parentheses).
  • Increasing from 0 to 3:
  • Interval: $(0, 3)$
  • Endpoints are open.
  • Constant from 3 to 28:
  • Interval: $(3, 28)$
  • Endpoints are open.
• Important conventions mentioned:
  • Endpoints in these largest open intervals are always written with parentheses:
  • For example, on the interval from 0 to 3 that is increasing: $(0,\,3)$.
  • Domain versus range:
  • Domain concerns the x-values (where the function is defined).
  • The interval notation for monotonic pieces uses x-values.
• Extra examples from the discussion:
  • Another portion of the discussion read: increasing on $(-8, -6)$, decreasing on $(-6, 0)$, increasing on $(0, 6)$, and then decreasing after 6. These are illustrative pieces showing how a function can switch monotonicity across different x-intervals.
• Domain and range from the practice example:
  • Domain is the union of all x-intervals on which the function is defined; using the intervals above, a possible domain structure could be
  • $(-\infty, 0) \cup (0, 3) \cup (3, 28)$
  • Range is the set of possible y-values corresponding to those x-intervals. In the example, the lowest y is 0 and the highest discussed value is 2, giving something like
  • $\text{Range} = [0, 2]$
  • Note: If the graph continues beyond the shown intervals in the real problem, the range can extend further (e.g., toward $\infty$) depending on the graph’s behavior beyond 28.
• Important caution about holes and discontinuities:
  • If the graph is not continuous on an interval, we call that a discontinuity.
  • A common type shown is a hole in the graph, which creates a missing point in the domain. The domain should reflect any points where the function is not defined.

Continuity and discontinuities: holes, breaks

• Discontinuity (hole) in a graph:
  • A hole represents a point where the function would be defined by a simple rule, but is not defined in the given graph, creating a break in the domain.
  • When a graph has a hole, you should note the x-value at which the hole occurs and exclude that point from the domain.
• Continuity assessment:
  • A function is continuous on an interval if there are no breaks, holes, or jumps in that interval.
  • If a graph has a hole, it is not continuous at the x-value where the hole occurs.

Cube root function: monotonicity and continuity

• Function considered: the cube root function $f(x) = \sqrt[3]{x}$
• Key evaluation example:
  • For input $x = -8$, the cube root is $\sqrt[3]{-8} = -2$ since $(-2)^3 = -8$.
• Monotonicity:
  • The cube root function is increasing on all of $\mathbb{R}$; it is never decreasing and never constant on any interval.
• Continuity:
  • The cube root function is continuous for all real numbers; no discontinuities occur.
• Reading from the graph or as a rule, you can plot points like (-8, -2), (0, 0), (8, 2) and observe a smooth, continuous, strictly increasing curve.

Square root function: domain, range, and monotonicity

• Function considered: the square root function $f(x) = \sqrt{x}$
• Domain considerations:
  • The square root is defined only for nonnegative inputs: $x \ge 0$.
  • Therefore, the domain cannot include negative numbers. In interval notation, the domain is typically
  • $[0, \infty)$ (for a graph that continues indefinitely to the right).
  • If a graph is shown only up to a finite right endpoint, say $x \le 8$, then the domain would be
  • $[0, 8]$ (the endpoint 8 included if the graph reaches it).
• Endpoints and inclusion:
  • Since \sqrt{0} = 0, the left endpoint x = 0 is included, so the domain begins with a bracket: $[0, \dots)$ or $[0, 8]$ depending on the graph extent.
• Range considerations:
  • As x increases from 0, $f(x) = \sqrt{x}$ increases as well. If the domain ends at x = x{max}, then the range ends at $f(x</em>{max}) = \sqrt{x<em>{max}}$. For example, if the shown domain ends at x{max}=8, then the range is
  • $[0, \sqrt{8}] = [0, 2\sqrt{2}] \approx [0, 2.828]$.
• Monotonicity:
  • On its domain, $f(x)=\sqrt{x}$ is increasing (strictly) for all x > 0; there is no decrease or constant segment.
• Common pitfall illustrated in the transcript:
  • When evaluating expressions with negative numbers, be careful with calculator input. For example, to square -2 correctly you must type $(-2)^2 = 4$ and not $-2^2 = -4$ (the latter is interpreted as $-(2^2) = -4$).

Practical reminders and study tips

• Always identify whether you are describing intervals in terms of x-values (domain) or y-values (range). The standard convention uses x-values for intervals of monotonicity.
• Use parentheses for open intervals when describing largest open intervals of monotonicity or constant behavior:
  • Decreasing on $(-\infty, a)$, Increasing on $(a, b)$, Constant on $(b, c)$, etc.
• Use brackets for domain endpoints that are included in the domain (closed endpoints):
  • Example: $\text{Domain} = [0, \infty)$ or $[0, 8]$ depending on the graph.
• When a graph has a hole or discontinuity, note the x-value of the hole and exclude it from the domain; the function is not continuous at that point.
• For cube roots, remember the inverse property: if you know a value y, then $f(x) = \sqrt[3]{x}$ implies $x = y^3$; for instance, $\sqrt[3]{-8} = -2$ because $(-2)^3 = -8$.
• Always cross-check monotonicity by selecting representative intervals and verifying the sign of the derivative is not required here; the visual (or tabular) evidence should align with the interval notation you produce.
• Practice problems suggested by the transcript:
  • Given a table, identify f(0) and f(6).
  • Read off f(0), f(3) from a graph, and determine whether the function is constant over a segment.
  • Determine the largest open intervals on which a function is increasing, decreasing, or constant.
  • Identify domain and range for square root and cube root graphs, noting any holes or discontinuities.
  • Be mindful of calculator input for negative numbers and exponents.

Quick reference formulas and notation

• Function notation: $f(x)$, with values determined by inputs: if the table lists $(xi, yi)$, then for that input $f(x<em>i)=y</em>i$.
• Monotonicity intervals (example conventions):
  • Decreasing: $(-\infty, a)$
  • Increasing: $(a, b)$
  • Constant: $(b, c)$
• Key function domains:
  • Square root: $\text{Domain} = [0, \infty)$
  • Cube root: $f(x) = \sqrt[3]{x},\quad x \in (-\infty, \infty)$
• Sample calculations:
  • $\sqrt[3]{-8} = -2\quad (\text{since } (-2)^3 = -8)$
  • $(-2)^2 = 4\quad (\text{note the need for parentheses})$

End of notes