Functions: further domain and range, modulus function, sketching graphs

Domain and Range – The “Ticket Rules” 🎟️

Think of the domain as the list of people who can enter a concert. Only those who meet the criteria (the “ticket rules”) are allowed. The range is the set of all possible reactions (the music) that the audience can experience.

Finding the Domain

1. Identify the basic rule for each type of function:
   • Rational: denominator ≠ 0
   • Exponential: always defined (except for some bases)
   • Logarithmic: argument > 0
   • Trigonometric: depends on the function (e.g., tan x undefined at π/2 + kπ)
2. Check for extraneous restrictions (e.g., square roots require non‑negative radicands).
3. Combine all restrictions using set notation or interval notation.

Finding the Range

Once the domain is known, evaluate the function’s output possibilities. For many functions, the range is all real numbers, but for others it may be bounded.

Example: Rational Function

Consider $f(x)=\dfrac{x^2-4}{x-2}$.

• Domain: $x eq 2$ → $(-\infty,2)\cup(2,\infty)$.
• Factor numerator: $(x-2)(x+2)$.
• Cancel: $f(x)=x+2$ for $xeq2$.
• Range: $(-\infty,\infty)$ (since $x+2$ covers all reals).

Domain & Range Table

Modulus Function – The Mirror Mirror 🪞

The modulus (or absolute value) function turns every number into its distance from zero, never negative. It’s like a mirror that reflects negative numbers to the positive side.

Definition

$|x| = \begin{cases} x, & x \ge 0\\ -x, & x < 0 \end{cases}$

Key Properties

• Even function: $|x| = |-x|$ – symmetric about the y‑axis.
• Graph is a “V” shape with vertex at (0,0).
• Always non‑negative: $|x|\ge0$.

Example: $f(x)=|x-3|$

Shift the basic V‑shape right by 3 units.

• Piecewise: $f(x)=\begin{cases} x-3, & x\ge3\\ -(x-3), & x<3 \end{cases}$
• Vertex at (3,0).
• Symmetry about the line $x=3$.

Graph Sketch (no image, use emoji)

Imagine a V‑shape made of two straight lines meeting at (3,0). The left side slopes up to the left, the right side slopes up to the right.

Sketching Graphs – Step‑by‑Step Guide 🚀

Sketching a graph is like planning a road trip: you need a map (domain), stops (intercepts), and landmarks (asymptotes).

General Steps

1. Domain – list all allowed x‑values.
2. Intercepts – find where the graph crosses the axes.
3. Symmetry – check if the function is even, odd, or neither.
4. Asymptotes – vertical, horizontal, or oblique.
5. Behaviour near asymptotes – does the graph go to ±∞?
6. Plot key points – choose values within the domain and calculate y.
7. Sketch – connect points smoothly, respecting asymptotes and symmetry.

Example 1: $y=\dfrac{1}{x-2}$

• Domain: $xeq2$.
• Vertical asymptote: $x=2$.
• No y‑intercept (x=0 → y=−½).
• Horizontal asymptote: none (approaches 0 as $x\to\pm\infty$).
• Plot points: $x=0\to y=-½$, $x=1\to y=-1$, $x=3\to y=1$, $x=4\to y=0.5$.
• Sketch two branches: one in Quadrant II (negative y) approaching $x=2$ from the left, and one in Quadrant I (positive y) approaching $x=2$ from the right.

Example 2: $y=|x^2-4|$

• Start with $x^2-4$: zeros at $x=\pm2$.
• Graph of $x^2-4$ is a parabola opening up, crossing the x‑axis at ±2.
• Apply modulus: reflect any negative parts above the x‑axis.
• Result: a “W” shape with vertices at (−2,0), (0,4), and (2,0).
• Symmetry: even function (mirror about y‑axis).

Quick Tips

• Use a table of points to keep track of calculations.
• Check end behaviour to decide if the graph flattens out.
• Remember that modulus flips negative y‑values to positive.
• When in doubt, plot more points to confirm the shape.

Practice Problems – Try Them Out! 💡

1. Find the domain and range of $f(x)=\dfrac{x+3}{x^2-1}$.
2. Sketch $g(x)=|x+5|$ and identify its vertex.
3. Sketch $h(x)=\dfrac{x^2-9}{x-3}$ and state any asymptotes.
4. Determine if $k(x)=|x^3-1|$ is even, odd, or neither.