Rational Functions, Piecewise Graphing, and Average Rate of Change

Determining Domains of Rational and Radical Functions

To find the domain of a rational function such as $\frac{1}{3x^2 - 6x}$, the denominator must be set as not equal to zero ($3x^2 - 6x \neq 0$). Solving this quadratic expression by factoring out the greatest common factor results in $3x(x - 2) \neq 0$, which yields exclusions at $x \neq 0$ and $x \neq 2$. In interval notation, the domain is $(-\infty, 0) \cup (0, 2) \cup (2, \infty)$. For radical functions like $f(x) = \sqrt{x + 2}$, the radicand must be non-negative ($x + 2 \geq 0$), resulting in a domain of $[-2, \infty)$. Standard square roots require non-negative results to remain within the set of real numbers, as negative radicands produce imaginary values involving $i$, defined as $\sqrt{-1}$. Radicals can be separated by multiplication or division but cannot be separated over addition or subtraction.

Combined Restrictions in Domain Analysis

When a radical exists in the denominator, such as $\frac{x}{\sqrt{6 - x}}$, the conditions for rational and radical functions must be merged. While a radical typically allows for values greater than or equal to zero, being in the denominator restricts it to being strictly greater than zero ($6 - x > 0$). Solving this involves subtracting $6$ and dividing by $-1$, which requires flipping the inequality symbol to yield $x < 6$, or $(-\infty, 6)$. In cases with multiple restrictions, such as $\frac{\sqrt{x + 2}}{-5 + x}$, each part is solved individually ($x \geq -2$ and $x \neq 5$) and combined into the interval notation $[-2, 5) \cup (5, \infty)$.

Graphical Analysis of Domain and Range

Domain and range can be determined visually from graphs by observing the horizontal and vertical spans. For a standard parabola opening downward with a vertex at $(0, 4)$, the domain is $(-\infty, \infty)$ while the range is $(-\infty, 4]$. On more complex graphs, such as a segment starting with an open circle at $(-6, 0)$ and ending within a specific range, brackets denote inclusive points while parentheses signify open boundaries. For a graph spanning from $x = -6$ to $x = 0$ with a lowest y-value of $-3$ (inclusive) and a highest y-value of $4$ (non-inclusive), the domain is $(-6, 0)$ and the range is $[-3, 4)$.

Piecewise Functions and Graphing Boundaries

Piecewise functions consist of multiple function segments, each defined over a specific, non-overlapping domain boundary. Graphing these requires creating separate tables of values for each piece and carefully marking boundary points as open or closed circles. If a function is defined as $2x - 7$ for $x \leq 3$ and $5$ for $x > 3$, the value at $x = 3$ is a closed circle for the first piece and an open circle for the second. If an open circle and a closed circle overlap at the same coordinate, such as $(2, 4)$ in a three-part function, the closed point takes precedence, making the function continuous at that location.

Average Rate of Change and Function Behavior

The average rate of change measures how much a function increases or decreases over a specific interval of its domain ($[x_1, x_2]$). Unlike the constant slope of a linear function, the rate of change for non-linear functions varies depending on the interval chosen. The formula is expressed as the change in output over the change in input: $\frac{\Delta y}{\Delta x} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$. For the function $f(x) = x^2 + 5x$ on the interval $[3, 6]$, the outputs $f(3) = 24$ and $f(6) = 66$ are calculated first. The average rate of change is $\frac{66 - 24}{6 - 3} = \frac{42}{3} = 14$. On the interval $[-4, 1]$, the outputs are $f(-4) = -4$ and $f(1) = 6$, resulting in an average rate of change of $\frac{6 - (-4)}{1 - (-4)} = \frac{10}{5} = 2$.

Questions & Discussion

During the lecture, a student asked about the type of discontinuity occurring at $x = 5$ for the combined radical and rational function. The instructor noted it was a discontinuity but was uncertain if it characterized a jump or a hole without further graphing via Desmos. Another discussion involved set builder notation; the instructor clarified that while Lumen might include it, the course focuses primarily on interval notation. A question regarding the spacing of coordinate tick marks and the necessity of drawing graph lines to the edge of the coordinate plane was also addressed, with the instructor emphasizing that graphs should be drawn to the edge with arrows to indicate they continue infinitely.