Toolkit Functions, Domains & Ranges, Piecewise Functions, and Average Rate of Change

Domains and Ranges

• Domain: set of allowable inputs (x-values). Read from the x-axis; expressed in interval notation. Use brackets [] to indicate inclusion and parentheses () to indicate exclusion. Example: the domain could be $(-\infty, 3]$ or $(-\infty, 3)$ depending on whether 3 is included.

• Range: set of outputs (y-values). Read from the y-axis.

• Reading order: start with the smallest x-value and move left to right; include or exclude endpoints according to the bracket types.

• Endpoints and infinity: use $-\infty$ and $+\infty$ with parentheses; infinity endpoints are never included (always parentheses).

Reading from graphs

• To read domain: look at x-values that appear on the graph.

• To read range: look at y-values that appear on the graph.

• Interval notation basics:

  • If a and b are endpoints with a < b, the interval is written as $[a,b]$ (inclusive) or $(a,b)$ (exclusive) or mixed, e.g., $[a,b)$ or $(a,b]$ depending on inclusion.

• Holes or gaps create domain restrictions; the domain can be a union of intervals, e.g., $(-\infty, 0) \cup (2, \infty)$.

Toolkit functions: key functions and their domains/ranges

• Constant function: $f(x) = c$

  • Domain: $(-\infty, \infty)$

  • Range: {c}

  • Graph: horizontal line y = c

• Identity function: $f(x) = x$

  • Domain: $(-\infty, \infty)$

  • Range: $(-\infty, \infty)$

  • Graph: line y = x

• Absolute value: $f(x) = |x|$

  • Domain: $(-\infty, \infty)$

  • Range: $[0, \infty)$

  • Graph: V-shaped

• Piecewise function: e.g., $f(x) = \begin{cases} x, &amp; x \ge 0 \\ -x, &amp; x &lt; 0 \end{cases}$

  • Domain: usually all real numbers unless pieces restrict it

  • Range: depends on pieces (often $[0, \infty)$ for this example)

  • Notation: uses curly braces with conditions; can create domain gaps

• Polynomial functions (even vs odd powers):

  • Example: $f(x) = x^2$ and $f(x) = x^3$

  • Domain: $(-\infty, \infty)$ for both

  • Range: $x^2$ gives $[0, \infty)$; $x^3$ gives $(-\infty, \infty)$

• Reciprocal: $f(x) = \frac{1}{x}$

  • Domain: $(-\infty, 0) \cup (0, \infty)$ (exclude 0)

  • Range: $(-\infty, 0) \cup (0, \infty)$

• Reciprocal square: $f(x) = \frac{1}{x^2}$

  • Domain: $(-\infty, 0) \cup (0, \infty)$

  • Range: $(0, \infty)$

• Square root: $f(x) = \sqrt{x}$

  • Domain: $[0, \infty)$

  • Range: $[0, \infty)$

• Cube root: $f(x) = \sqrt[3]{x} = x^{1/3}$

  • Domain: $(-\infty, \infty)$

  • Range: $(-\infty, \infty)$

• Exponent-root relationships:

  • $x^{1/2} = \sqrt{x}$, $x^{1/3} = \sqrt[3]{x}$

  • Even roots have nonnegative outputs; odd roots cover all reals

Piecewise notation and domain pieces

• Piecewise notation: $f(x) = \begin{cases} \text{output 1}, &amp; \text{condition 1} \\ \text{output 2}, &amp; \text{condition 2} \ \cdots \end{cases}$

• Domain is the union of the domains of each piece; can include holes where no piece applies.

• Example intuition: if the piece says output is x for x >= 0 and -x for x < 0, the graph is the absolute value function; some intervals may be missing if no piece covers them.

Quadratics and Cubics (polynomials)

• Quadratic: $f(x) = x^2$

  • Domain: $(-\infty, \infty)$

  • Range: $[0, \infty)$

  • Graph: parabola opening up

• Cubic: $f(x) = x^3$

  • Domain: $(-\infty, \infty)$

  • Range: $(-\infty, \infty)$

  • Graph: S-shaped curve

Reciprocal-related functions and domains

• Reciprocal: $f(x) = \frac{1}{x}$

  • Domain: $(-\infty, 0) \cup (0, \infty)$

  • Range: $(-\infty, 0) \cup (0, \infty)$

• Reciprocal square: $f(x) = \frac{1}{x^2}$

  • Domain: $(-\infty, 0) \cup (0, \infty)$

  • Range: $(0, \infty)$

Even and odd roots

• Square root (even root): $f(x) = \sqrt{x}$ with $\text{Domain} = [0, \infty)$ and $\text{Range} = [0, \infty)$

• Cube root (odd root): $f(x) = \sqrt[3]{x}$ with $\text{Domain} = (-\infty, \infty)$ and $\text{Range} = (-\infty, \infty)$

How to read exponents and roots relationship

• $x^{1/2} = \sqrt{x}$, $x^{1/3} = \sqrt[3]{x}$

• Even powers produce nonnegative outputs for nonnegative inputs; odd powers preserve sign

Rates of change: average rate of change

• Definition: average rate of change of f on [a, b] is
  $\frac{f(b) - f(a)}{b - a}$

• This is the slope of the secant line through (a, f(a)) and (b, f(b))

• Meant to quantify how much the output changes per unit change in input over the interval

• Examples with f(x) = |x|:

  • a = 1, b = 5: $\frac{f(5) - f(1)}{5 - 1} = \frac{|5| - |1|}{4} = \frac{5 - 1}{4} = 1$

  • a = -1, b = 3: $\frac{f(3) - f(-1)}{3 - (-1)} = \frac{3 - 1}{4} = \frac{1}{2}$

• Interpretation:

  • If f is increasing on the interval, the average rate is positive

  • If f is decreasing, the average rate is negative

• Important distinction: instantaneous rate of change (limit as the interval shrinks) is a calculus topic to be covered later

• Practical note: when evaluating, ensure a ≠ b; if a = b, the expression is undefined (division by zero)

Quick context checks and notes

• Always check domain feasibility in word problems (e.g., pizza cost, physical constraints)

• For reading graphs, identify the axes: x-axis for domain, y-axis for range

• For solving, remember to use interval notation and remember which endpoints are included vs excluded

Next steps and instructor availability

• Instantaneous rate of change and limits are covered in Calc I later

• Office hours: Friday (check with instructor for exact times and appointment method)

• Bring questions and any assignment-specific graphs or functions to review before the next session